36

A CHARACTERIZATION OF FINITELY GENERATED MODULES by

Embed Size (px)

Citation preview

Page 1: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

������������������ ������� ������������������������������������! "����#�����$

%'&�(

)+*-,/. � *-,/. $1032'4�57615

�98/2:5<; =>;�;@? %BA =C8@8/5<6D=>EDF *-, 8 = *-.�G ? .CHB.>. A 5IE�8GKJL,MN,3* 61? * 8 = J EO4=P8/2Q� J E JN, ;�=CER * 8/2:5 A * 8 =S0I;

T 2:=P8 A * EU� JN.>. 5 M 5V7WNWLW

Page 2: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

��������� � �������������������������

��2:=S;=S;�8 J 0I5 , 8 = G & 8 2 * 88 2B5 * 0<0 J A F * E & =>E M 8 2:57;@=S; %'& )+*N, . � *-,/. $1032'4�576152 * ; % 5I5<E * 0<0 5<F18 576 =CEOF *-, 8 = *-.BG ? .CHB.C. A 5IE�8 J-G 8/2:5 , 5���?:= , 5 A 5IE�8/; GKJN,�MN,3* 6:?! * 8/= J EO4=C8 2Q� J E JL, ;�=>ED * 8 2B5 A * 8/=>0<;�"

#�* 8 , =S0 $ ) 5I5 G

T 2:=C8 A * EU� JL.C. 5 M 5 * & V<WLWNW

=>=

Page 3: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

����������� $

� E�8 ,/J 61?B0 8/= J E " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " V

$'? %BA J 61? . 57; J-G � =CE:=C8 5 . & ��5IE:5 ,/* 8 576O J 61? . 57;�" " " " " " " " " " " " " " " " " " " " " " " �

� � * 0 8 JN, =�� * 8 = J E J-G � =CE:=C8 5 . & ��5IE:5 ,/* 8 576O J 61? . 57; " " " " " " " " " " " " " " " " " " " " " VLV

��2:5 � * 0 8 JN, =�� * 8/= J E J-G � =CE:=C8 5 . & ��5IEB5 ,3* 8 576O J 6:? . 57;�=>;�# E:= ��?:5 " " " " " " " " " " " " " " " V �

� EQ��F:F . =>0 * 8 = J ED8 J # JL. & E J A = *N. ; " " " " " " " " " " " " " " " " " " " " " " " " " " " "����

� = % . = JNML,/* F:2 & " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " "�

=>=C=

Page 4: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

������������� ���������� ������������������ ����� �!�"�������� ��#%$&��#!'��(��)

*,+.-0/�1�2 ��3 /4-6587:9:;</�=>5 3

��2:5 F:? , F J ; 5 JNG 8/2:=>;�8/2:5<; =S; =S;�8 J 032 *N,/* 0 8/5 , =��<5 H E:=C8 5 . & M 5IEB5 ,3* 8 576 A J 61? . 5<;�" � ;@8 * E 6 *-, 6 , 5<; ? . 8+=>E

?:EB6:5 ,/MN,3* 61? * 8 5 * % ;@8 ,3* 0 8 *-.>M 5 % ,/* =S;�8 2 * 8�5�?N5 , & H E:=C8 5 � % 5 . = * E MN,/J ?BFQ=>;�=S; J A JL, FB2:=>0+8 JO* ?:E:= ��?:5 6:= , 5<0 8

F ,/J 61?B0 8 J-G 0 & 0 . =S0 ML, J ?:FB; J-G F , = A 5OF J 4�5 , JL, 6:5 , "�� ;@= A = .S*-, ;�8 * 8/5 A 5IE�8 0 * E % 5 A * 615 * % J ?18 H E:=C8 5 . &

M 5<E:5 ,3* 8/5<6 MN,/J ?BFB;4@�5�A:0 5<F18 =>E 8 2:=S; 0 * ; 5 8 2:5 , 5 *N, 5 *N. ; J =CE H E:=C8 5O0 & 0 . =>0 ; ? A A * EB6:;�"R��2:=>; , 57;@? . 8 0 * E % 5

M 5<E:5 ,3*-. = �I5<6 G ? , 8 2:5 , 8 JCB A J 61? . 57;�42:5 , 5 B =S; * � ?B0 . =S615 * E 6 J A * =>E " ��2:=S;+=S;�8 2:5 A * =CE , 5<; ? . 8 JNG 8/2:=>;

8 2B5<; =>;�" ��2:5 , 5 *-, 5 *-. ; J ; J A 5 J 8 2:5 , ;@8 * 8 5 A 5<E�8/; * % J ?18 J 8/2:5 , A JN, 5 M 5IE:5 ,/*N. = �I576 6 J A * =CE ; * ; 4�5 .>. " ��2:=>;

8 2B5<; =>;4= .>. 0 J E 0 . ?B615+4=C8 2 * E * F:F . =S0 * 8 = J EO8 JD�E F.G A J 61? . 5<;�"

� 5 8?B;615 H E:5 * ER� % 5 . = * E ML, J ?:F * EB6 * , =CE M * ;�E JN, A *N.C. & 6 J E:5IH � *-.>. = * E � V @ � �KJML�" T 5�4= .>. 0 *-.>. 8 2B5

=S615IE�8 =C8 & 5 . 5 A 5IE�8ON " � 5I8 ?B;615 H E:5 * E B A J 61? . 5 * ; GKJL.C.>J 4 ; (

#!�"���4���� ���� 1�PQ1 ��E B A J 6:? . 5R=S; * E � % 5 . = * E MN,/J ?:F ?:EB615 , * 6:6:=P8/= J E8@�4=C8 2 ;/0 *-.S*-, A ? .

8 =>F . =>0 * 8/= J E GK,/J A * 0 J A A ?:8 * 8/=R?L5 ?:EB=P8 *-, & , =>E MSB @ * EB6 GKJN, * E &UT�VXWCY[Z * EB6]\ VX^_Y B 4�5O2 * ?N5 8 2 * 8

\`H Tba]W Ldce\ Tba \ W @fHQ\ a]^ L T ce\ Tba]^0T @8Hg\ ^ L T ch\iH ^0T L�@ * EB6 V T c T "

� J 8 =S0 5R8/2 * 8 8/2:=S; *-. ; J = A F . =>5<; 8 2 * 8�N T cjN ;@=>EB0 5CN a N T cjN T ckHQN a NML T cjN Tla N T @�; J %'&

0 * EB0 5 .C.S* 8/= J E8@mN�c&N T @ * EB6";@= A = .S*-,/. & 8 2:=S;�= A F . =C57;�8 2 * 8 GKJL,+* E & \ Y B @m\6N!c�N " T 5 4= .>. % 5 .>J'J $'=>E M =>E

F *N, 8/=>0I? .>*N, * 8 42 * 8 2 * F:F 5IEB; 42:5IE 8 2:5 , =>E M =S; * � ? 0 . =>615 * E � J A * =>E "Q� �?B0 . =S615 * E 6 J A * =>EhH � *N.C. = * E

V � L�n =S; * E =CE�8 5 MN,3*-. 6 J A * =>E 42:5 , 5 8 2:5 , 5 =>; *DG ?:EB0 8/= J E]o GK,/J A 8 2B5 E J E �I5 ,/J 5 . 5 A 5<EL83; JNG n 8 J 8 2B5

E J E! EB5 ML* 8 =p?N5�=>EL8/5 M 5 , ;;@?B032D8/2 * 8 GKJL,�*-.>. E J E! �I5 ,/J T�VXW�Y n�@qo�H T L,reosH TiW Lt@ * EB6 GKJN,�*N.C. TsVuW�Y n 42:5 , 5 W

Page 5: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

=S; E J E! �<5 ,/J @18/2:5 , 5 54A'=S;@8�5 . 5 A 5IE�83;�� * EB6 \ =>E n ; ?B032R8 2 * 8 T c W � a \i@ 42B5 , 5b\Ic N JN, osHQ\6L�� o�H W L�"��

=S; * � ? 0 . =>615 * ED6 J A * =>E 4=C8 2 o�H F L(c�� F � * EB6 ; J =S; D�E FqG 42:5 , 5 D =>; * H 5 . 6 * EB6�o�H� H F L L(c o� �mH� H F L L " �

� ? 0 . =>615 * ER6 J A * =CER=S; * F , =>EB0I=CF *N. =S615 *N. 6 J A * =CE * EB6 *N. ; J 8 2'?B; * ?:EB= ��?:5 G * 0 8 JL, = � * 8 = J ER6 J A * =CE H � *N.C. = * E

V7W L "

#!�"���4���� ���� 1sP�� T 5 0 *-.>. * E J E! 5 A F18 & ; ? % ; 5 8 J-G* E B A J 61? . 5�� * ; ? %:A J 61? . 5 = G =C8 =S; *

; ? % MN,/J ?BF ?:EB6:5 ,�* 6:61=C8 = J E * EB6 *-. ; J =S; 0 .>J ; 5<6O?:EB6:5 , A ? . 8/=CF . =S0 * 8/= J E GK, J A 8 2:5 , =>E M�B "

� J 8 =S0 5 8 2 * 8�4�5 0 * EU8 2:=>E $ JNG � % 5 . = * E ML, J ?:FB; * ;�� A J 61? . 5<;�"��2 * 8�=>; 8 J ; * & GKJN,�* E &�TCYUZ @ * EB6

; J A 5�=>E�8 5 M 5 ,�� @ � T�� ? ;�8 A 5 * E ; T�aST a���� �ua T @ � 8 = A 5<;�@'4=P8/2�N T % 5I=>E M 8/2:5�=>615<E�8 =C8 & 5 . 5 A 5IE�8�=>E Z "�� 8

=S;0 . 5 *-, 8/2 * 8 A ? . 8/=CF . =S0 * 8/= J E GK,/J A 8/2:5�=CE�8/5 M 5 , ;�=S;61=S;�8 , = % ?18/=R?L5 = G * EB6 J E . & = G Z =>;� % 5 . = * E " � JN, ; ?:F:F J ; 5

A ? . 8/=CF . =S0 * 8/= J EQ=>; 61=S;�8 , = % ?18/=R?L5K@:8/2:5IE H T a]W L a H T a W L(c �.H T�a W L c � Tba � W ; J Wda]T c T�a W "�� JL, 8 2B5

0 J Ei?N5 , ; 5 ; ?:F:F J ; 5�8 2 * 8 J ? ,�MN,/J ?BFO=S;� % 5 . = * E8@18/2:5IE��dH T�a W Ldc H T�a W L a���� �0a H T�a W Ldc�� T�a � W "

#!�"���4���� ���� 1�P�� � 5 8�� % 5 * E B A J 61? . 5 * EB6 . 5 8 T! 6V T#"<V $�$�$�V T&%O% 5 5 . 5 A 5IE�8/; J-G � " T 5

4= .C. 6:5 H E:5(' T V�$ $�$�V T %*) c�+-, T �/. T V $�$�$�V T %�0 c1.0N a \ T a���� � a \ % T % � \ V�$�$ $4V \ % Y B20 "

� J 8 5�8/2 * 8�+-, T �dH3ML c1.0N 0 "

���b�����"$ 1sP>1 ' T! <V�$ $�$4VXT&% ) =S; * ; ? %:A J 61? . 5 J-G � " *�-658554X2 $'?:F:F J ; 5 B =>; * , =>E M @�8/2 * 86�

=S; * E B A J 6:? . 5M@ * EB6 8/2 * 8�. T! 0V�$ $�$4VXT&% 0Q*N, 5D5 . 5 A 5<EL83; J-G � " � J 47' T! 6V�$�$ $4VXT#% ) =S; E J E! 5 A F18 & ;@=>EB0I5

N a V T& �a8� ��� a V T&% c T! ma8� ��� a�T&%�Y ' T! 6V $�$�$�V T&% ) "���2:5<E 8 * $N5 * E &IT�VXW,Y ' T! 6V $�$ $�V T&% ) @ %'& 8 2:5�615 H E:=P8/= J E

J-G ' T V $�$ $�V T %9) 4�5 $'E J 4 8/2 * 8 T c \ T a:� ���`a \ % T % * EB6 W c ^ T a:� ���`a[^ % T % @�42:5 , 5�\<; * EB6 ^ ;

*-, 5 , =>E M 5 . 5 A 5<E�8/;�@ GKJN,�= c V V�$�$ $�V?> " ��2:5IE TA@eW c Hg\ @h^ L T a���� �ia Hg\ % @ ^ % L T % Y Z * EB6 *N. ; J

\`H T L�c�Hg\ \ L T a�� ���0a Hg\ \ % L T % Y ' T V $�$�$�V T %*) "���2�? ;B' T V $�$�$�V T %9) =S; * ;@? %BA J 61? . 5 JNG � "

#!�"���4���� ���� 1sP�C T 5O0 *-.>.�* E B A J 61? . 5K@D� @ DE � ��� �GFIH���E�������� �KJ = G � 2 * ; * H E:=C8 5R;@? % ;@5I8

. T! 6V�$�$ $�V T&% 0 ; ?B032O8 2 * 82' T! 6V $�$ $�V T&% ) cL� "

� ED8 2B=>;0 * ; 5�4�5+; * & 8/2 * 8B. T V�$ $�$�VXT %M0+M 5IEB5 ,3* 8 57;4@1; F * E ;4@ JL, =>; * ; F * E:EB=CE M ; 5 8 GKJN, � " T 5+; * & 8 2 * 8

� =>; 0 & 0 . =S0+= G 8/2:5 , 5 54A'=S;@8/; * ; =>E MN. 5 5 . 5 A 5<EL8 T!Y � ;@?B032R8/2 * 8N' T ) cO� " � J 8/=>0I5+8 2 * 8 = G 4�5 0 J E ;@=S615 , *

?:E:=C8 *N, & , =>E M * ; * A J 61? . 5 J ?N5 , =C8/; 5 .CG @'8 2:5 A J 61? . 5�=S; H E:=C8 5 . & M 5IEB5 ,3* 8 576m@ * EB6O54?L5IEO0 & 0 . =>0�;@=>EB0I5�' V ) c B "

� G 4�5+0 J EB;@=S615 , B�P B * ; *lB A J 6:? . 5+=C8�8 J'J =>; H EB=P8/5 . & M 5<E:5 ,3* 8/5<6 4=C8 2 * M 5<E:5 ,3* 8/=CE M ;@5I8Q. H V V N L V H>N V V L 0 "

Page 6: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

�:? , 8/2:5 , A JN, 5K@ = G 4�5 . 5 8 B % 5 * A J 61? . 5 J ?N5 , =C8/; 5 .CG @�8 2:5 ;@? %BA J 61? . 57; JNG(B�*N, 5 5�A * 0 8 . & 8/2:5 =>615 *-. ; JNG�B @

; =CEB0I5�8 2:5+=S615 *N. ; *N, 5�8/2:5 ;@? % MN,/J ?:F ; JNG B 8 2 * 8 *-, 5�0 .CJ ;@576 ?BEB615 , A ? . 8 =>F . =S0 * 8 = J E %'& *-.>. 5 . 5 A 5<EL83; J-G B "

T 5 0 * E *N. ; J 0 J E ;@=S615 , ; J A 5�5�A * A F . 5<; H E:=P8/5 . & M 5IE:5 ,/* 8 5<6 MN,/J ?BFB; * EB6 A J 6:? . 57;�"

��� +������� 1�PQ1 ��E & H E:=C8 5 A J 61? . 5+=S; H E:=C8 5 . & M 5<E:5 ,3* 8/5<6 "�$1= A F . & 8 * $N5�8/2:5 M 5IE:5 ,/* 8 =>E M ; 5 8�8 J % 5

54?L5 , & 8 2:=>E M =CED8 2B5 A J 61? . 5�"

��� +������� 1�P � ��E & 0 & 0 . =S0 A J 61? . 5�H * EB6 *N. ; J 0 & 0 . =S0 MN,/J ?:FsL�=S; H E:=C8 5 . & M 5<E:5 ,3* 8/5<6O4=P8/2R8 2:5 M 5IE

5 ,3* 8 JN,�J-G 8 2:5 0 & 0 . =>0 A J 61? . 5 % 5<=CE M 8/2:5 ; F * E:EB=CE M ; 5 8 GKJL, 8 2:5 A J 61? . 5 "

��� +�������S1sP � ��2:5 ,3* 8 = J E *-. E�? A % 5 , ; *N, 5 E J 8 *+H E:=P8/5 . & M 5<E:5 ,3* 8/5<6(� A J 61? . 5�?:E 615 , 8 2B5 J F 5 ,/*

8 = J E a� =>E J 8/2:5 , 4 JL, 6:; 8 2:5 & *N, 5�E J 8 *�H E:=C8 5 . & M 5IE:5 ,/* 8 5<6 MN,/J ?BF " ��;/; ? A 5�8 2 * 8 8 2B5�; 5 8 .?, � � V $�$ $4V , %�� � %�0

=S; * M 5IEB5 ,3* 8 =>E M ; 5 8 GKJN,�� " � 5 8 � % 5 8 2B5 . 5 * ;�8�0 J A A J E A ? . 8 =>F . 5 JNG 8 2:5�615IE J A =CE * 8 JL, ; J-G J ? ,�M 5IE:5 ,/* 8 =>E M

; 5 8�@ 8/2 * 8�=>; . 5 8���c���� � H� 6V�$�$ $4V � % L "+� G 0 J ? , ;@5�� A ?B;@8 % 5 E J E �<5 ,/J " � J 8/=>0I5 8 2 * 8 * E & . =>E:5 *N, 0 J A % =>E *

8 = J E J-G 8 2B5 5 . 5 A 5<EL83;�=>EQ8/2:5 M 5IE:5 ,/* 8 =>E M ; 5 8�@ ; * & F���� @ 4= .>. 2 * ?N5 * 6:5IE J A =>E * 8 JN, 8 2 * 8�6:=R?'=S615<;�� * ; .CJ E M

* ; F * EB6 � *-, 5 , 5 .S* 8 =p?N5 . & F , = A 5�" � ?:8 8/2:=>; =S; E J 8 * F ,/J % . 5 A ; =>EB0 5D54?L5 , & E J E! �I5 , J ,3* 8/= J E *-. 0 * E % 5

4 , =P8 8 5IEQ=>ER8 2 * 8 GKJL, A "���2B5IE8@�0 J EB;@=S615 , 8 2:5 ,3* 8/= J E *-. E�? A % 5 , V � H � a V L "�$'=>EB0I52� a V 6 J 57;E J 8 61=p?�=S6152�i@

V � H � a V L �Y ' , �� � V�$ $�$4V , %�� � %9)�* EB6 8/2:5 , 5 GKJL, 5M@'8 2:5 ,3* 8/= J E *-. E'? A % 5 , ; *N, 5�E J 8 H E:=P8/5 . & M 5IE:5 ,/* 8 5<6 "

T 5 0 * ED615 H E:5 * 2 J A J A JN, F:2:=S; A % 5 8�4�5I5<E 8�4 J�B A J 61? . 5<; * ; GKJN.>.CJ 4 ;�"

#!�"���4���� ���� 1�P�� � 5 8�� % 5 * A J 6:? . 5 J ?L5 , * , =CE MB�* EB6 . 5I8�� % 5 * A J 61? . 5 J ?L5 ,OB�*-. ; J @

8 2B5IE 4�5 0 *-.>.�* A * FBF:=CE M�� ( ��� � * 2 J A J A JL, FB2:=>; A = G�GKJL, * E & � * EB6I� =>E � * EB6 GKJL, * E & \ =CE B @

� H � a �fLdc � H � L a � H �fL * EB6 � Hg\ � L(ch\ � H � L "

� J 8 =S0 5 8 2 * 8O42:5<E B =S; * ?:E:=C8 *-, & , =>E M @ � =S; * 2 J A J A JL, F:2B=>; A = G * EB6 J E . & = G�� Hg\ � a \ �fLCc

\ � H � L a \ � H �fL�"

#!�"���4���� ���� 1�P"! T 5 0 *-.>. * % = � 5<0 8/=R?L5+2 J A J A JL, FB2:=>; A * ED=S; J A JL, FB2:=>; A "

���b�����"$ 1�P � � 2 J A J A JN, F:2:=S; A 42 J ; 5 6 J A * =CE =S; *RH E:=P8/5 . & M 5IE:5 ,/* 8 576 A J 61? . 5 =S; 0 J A

F . 5 8/5 . & 615I8 5 , A =CEB5<6 %'& =C8/; % 5I2 * ?�= JN, J E 8 2:5 M 5IE:5 ,/* 8 =>E M ;@5I8�" *�- 5f5 4X2 $'?:F:F J ; 5R8 2 * 8 4�5Q2 * ?N5 * 2 J

A J A JN, F:2:=S; A � % 5I8�4�5<5IE"8�4 J B A J 61? . 5<;B� * EB6$#U@ 42:5 , 5(� =S; H E:=C8 5 . & M 5IE:5 ,/* 8 5<6 4=P8/2 M 5IE:5 ,/* 8 =>E M

; 5 8 . T! V $�$�$4V T&% 0 " ��2:5<E 8 * $N5 * E &UTeY � ; J T c \ 4T! Oa���� � a \ %6T&% GKJL, ; J A 5!\ 6V�$�$ $4V \ %]Y B @�8 2'?B;

Page 7: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

� H T Lc � HQ\ �T! Oa������`a \ %6T&% Lc \ � H T& L a:��� �`a \ % � H T&% L�" $ J @ GKJL, * E & TeY � @ � H T L =>; 0 J A F . 5I8 5 . &

615I8 5 , A =CEB5<6 %'& 8 2:5+= A *-M 5 JNG 8/2:5 M 5<E:5 ,3* 8/=CE M ; 5 8 "

* E & J-G 8 2:5 G * A = . = *N, F , J F 5 , 8/=C57; J-G 2 J A J A JL, FB2:=>; A ;�2 JN. 6 " � JL, 5�A * A F . 5K@

���b�����"$ 1sP � $'?:F:F J ; 5�8 2 * 8 � =>; * E B A J 61? . 5�2 J A J A JN, F:2:=S; A % 5 8�4�5I5<E�8 2:5 8�4 JOB A J 61? . 57;

� * EB6�� @ * E 6 8/2 * 8�� =>; * ; ? %:A J 61? . 5 J-G � "��:? , 8 2B5 , ; ?:F:F J ; 5 8/2 * 8�� =S; *�B A J 6:? . 5�2 J A J A JN, F:2:=S; A

% 5 8�4�5I5IE � * E 6 # "

� +�� � H � L�=>; * ;@? %:A J 6:? . 5 JNG � *�-658554 2 � JN,+* E & � H � L V � H �fL Y � H� L�@ 4�5 2 * ?N5 8 2 * 8 � H � L @

� H �fL�c � H � @ �fL Y � H� L�@ * EB6 *-. ; JOGKJL, * E & \ Y B @:\ � H � L�c � Hg\ � L Y � H� L�"R��2:5 , 5 GKJN, 5 � H � L�=>;

0 .>J ; 5<6O?:EB6:5 , A ? . 8/=CF . =S0 * 8/= J E GK, J A B "

��� � � H��(L�=S; * ; ? %:A J 61? . 5 J-G � *�-6585 4X2 ��2:5 G * 0 8 8 2 * 8 � H��(L�=S; E J E! 5 A F18 & * EB6 0 .>J ; 5<6 ?BEB615 ,

MN,/J ?BF J F 5 ,/* 8 = J EB; G *N.C. ;+= A A 5761= * 8 5 . & GK,/J A 8/2:5 G * 0 8 8/2 * 8 �� H � L�=S; * ; ? % MN,/J ?BF JNG � "D� J 4 0 J EB;@=S615 ,

A ? . 8/=CF . =S0 * 8/= J E GK,/J A 8 2:5 , =CE M"B " � 5I8 � Y �� H � L * E 6b\ Y B "���2B5IE %'& 8 2:5�615 H E:=C8 = J E JNG �� @ � H � L Y �O@

; J 8 2:5<E�\ � H � L Y ��; =CE 0 5���=>; * ; ? %:A J 61? . 5 JNG * E J 8 2:5 ,(B A J 61? . 5 "���2'?B; � HQ\ � L Y � * EB6�\ � Y � H � L

42:=S032Q0 J A F . 5 8/5<;�8/2:5+F ,/J�JNG "

� ;�� ��2:5 0 J A F J ;@=C8 = J E J-G 8�4 J 2 J A J A JN, F:2:=S; A =S; * 2 J A J A JN, F:2:=S; A " *�-658554 2+T 5+4= .C. ; 2 J 4 8 2 * 8

� 0 J A F J ; 5<6 4=P8/2�� =>; * E B A J 6:? . 5 2 J A J A JN, F:2:=S; A % 5I8�4�5<5IE � * EB6$# " � JL, 8 * $N5 * E & \ Y B * EB6

F V � Y � @'8 2B5IE � H��"HQ\`H F a � L Ldc � Hg\��"H F L a \���H � L L�ch\ � H��"H F L L a \`H � H���H F L L�"

� 7�� ��2B5"=>Ei?N5 , ; 5 G ?:EB0 8 = J E J-G * E =>; J A JL, F:2B=>; A =>; * =S; J A JN, F:2:=S; A " *�-658554 2 ��2:=S;D=>;O5 * ; & 8 J

; 5I5 GKJL, ; ?:F:F J ; 5 � =>; * E =S; J A JN, F:2:=S; A % 5I8�4�5<5IE 8�4 J B A J 61? . 5<;�@ � * EB6 �]@ 8 2:5<E �� =S; * % = � 5<0 8/= J E

% 5 8�4�5I5IE � * EB6�� "9� J F ,/J ?N5D8/2 * 8 =C8 =S; * E * 6:61=C8 =p?N5Q2 J A J A JN, F:2:=S; A 0 J EB;@=S615 , � H F a � L 42:5 , 5

F V � Y # "�� J 8 =S0 5 8 2 * 8 � H �� H F a � L Ldc F a � c � H � H F L L a � H �� H � L L�c � H �� H F L a �� H � L L�" ��2'?B;

� H F a � Ldc � H F L a � ; =>EB0 5 � =>; J E:5 8 J J E:5 " � J 4 8 J F , J ?L58 2 * 8 � HQ\ F L(ch\ � H F L "� J 8 5 8 2 * 8

� H �� Hg\ F L L"c \ F c \ � H �� H F L L,c � Hg\ �� H F L L " $ JO*NML* =CE �� Hg\ F L,c \ �� H F L 42B=>032"F , J ?L5<;�8/2 * 8�8 2B5

=>E ?L5 , ;@5 G ?:EB0 8 = J E J-G�* ED=S; J A JL, FB2:=>; A =S; * ED=>; J A JL, F:2B=>; A "

� ��� ��2:5 $N5 , E:5 . J-G�* 2 J A J A JN, F:2:=S; A =S; * ; ? %:A J 61? . 5 " *�-6585 4X2 ��2B=>;�=>;5 * ; & ; =CE 0 5 0 . 5 *N, . & . N 0 =>; *

; ? %:A J 61? . 5 JNG � " � J 8 =S0 5 8/2 * 8 � 0\ � c �� H .0N 0 L�; J 8/2:5IE ;@=>EB0I5 �� H .0N 0 L =S; * ;@? %BA J 61? . 5 JNG � @ 4�5

Page 8: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

*-, 5+6 J EB5 "

T 5�4= .C. E J 4 F , 5<; 5IE�8�; J A 5 G * A = . = *-,, 5<; ? . 8/; * % J ?18 A J 6:? . 57; * EB6 ; ? %:A J 61? . 5<;�" � E F *-, 8 =S0 ? .>*N, 4�5 *-, 5

=>EL8/5 , 5<;@8 576 =>E G * 0 8 JN, A J 61? . 5<;�"

���b�����"$ 1sP�C � � # GKJL, A ; * E B A J 6:? . 5�" *�-658554X2 � 5I8�� % 5 * E B A J 6:? . 5 * EB6 . 5 8 # % 5

* E & ; ? %:A J 61? . 5 " � 5 8 ?B;+6:5 H E:5 8/2:5 ; 5 8�� � # c . � a # � � Y � 0 " T 5 0 .>* = A 8/2 * 8�� � # =S; *N. ; JD* E

B A J 61? . 5 42:5 , 5�H �� a # L a H �I" a # L,c H � a �I" a # L * EB6 \iH �� a # L"c \ � a # " $'=>EB0 5 8 2B5

MN,/J ?BF =S; � % 5 . = * E8@ # =S; E JL, A *-. * EB6 8 2B5 HB, ;�8 J F 5 ,/* 8 = J E =>; 4�5 .C. 615 H EB5<6 * E 6 GKJL, A ; * MN,/J ?:F H � *N.C. = * E

V � ML " � J 4 0 J E ;@=S615 , 8/2:5U; 5<0 J EB6 J F 5 ,3* 8/= J E " T 5U4 * EL8 8 J F , J ?L5D8 2 * 8 A ? . 8 =>F . =>0 * 8 = J E GK,/J A 8 2:5 , =>E M

=S; 4�5 .C. 615 H E:576 " $'?:F:F J ; 5D8 2 * 8 � "a # c � "�a # @ 8 2'?B; � c � "�a � GKJN, ; J A 5 � Y # " ��2:5<E

\ � c \ � "ba ��� GKJN, ; J A 5���� Y # ;@=>EB0I5 # =S; 0 .CJ ;@576 ?BEB615 , A ? . 8 =>F . =>0 * 8 = J E GK,/J A 8 2:5 , =>E M " ��2'?B;

\ �� @ \ �I" c � �Y # * EB6 8/2:5 , 5 GKJL, 5 \ � a # c \ �I" a # "!��2:5 , 57;�8 J-G 8 2B5UF ,/J F 5 , 8 =>5<; GKJN.>.CJ 4

61= , 570 8 . & GK, J A 8/2:5 G * 0 8 8/2 * 8�� =S; * E B A J 61? . 5K@ % 5<0 * ?B; 5 GKJN,�* E &�T�VuW�Y � * EB6 \ Vu^IY B 4�5 2 * ?L5�8 2 * 8

\`H H Tma #SL a H WMa # L Ldce\iH T8alWKa # Ldc \ T8a \ WKa # c HQ\ Tma #SL a Hg\ WKa #SL ch\`H T8a # L a \iH WKa # L�@ * EB6 8 2 * 8

Hg\ a^ L�H T a #SL�c HQ\ a�^ L T a # c Hg\ T a # L a H ^0T a # Lt@ * EB6 *N. ; J 8 2 * 8�HQ\ ^ L4H T a # L(ce\ ^�T:a # ch\iH ^0T a # L

* E 6 H E *-.>. & @ V H T�a # Ldc V T�a # c T�a # " $ J � � # =S;�=C8/; 5 .CG * E B A J 6:? . 5�"

���b�����"$ 1sP � � G � * EB6 # *-, 5 A J 6:? . 57; J ?N5 , * , =>E M�B�* EB6 � =S; * E & ;@? , � 5<0 8 =p?N5+2 J A J A JN,

F:2:=S; A GK, J A � J E�8 J # 8/2:5IE = G � =>; H E:=C8 5 . & M 5IEB5 ,3* 8 576m@ ; J =S; # " *�-658554 2 � 5 8 B % 5 * , =>E M * EB6 . 5 8D� * EB6

# % 5 B A J 61? . 5<;�@ * E 6 *N. ; J ;@?BF:F J ;@5 8 2 * 8�� =S; H EB=P8/5 . & M 5IE:5 ,/* 8 5<6D4=C8 2 * M 5IE:5 ,/* 8 =>E M ;@5I86. T! 0V $�$�$�V T&% 0 "

T 5�4= .C. ;@2 J 4 8/2 * 8�8 2:5�;@5I8�. � H T! L V $�$ $4V � H T#% L 0�M 5IE:5 ,/* 8 57; # " � JL, 8 * $N5 * E & � Y #U@�; =>EB0 5 � =S;�;@? , � 5<0 8 =p?N5M@

8 2B5 , 5 5�A1=S;�83; � Y � ;@? 032�8 2 * 8 � H � L�c���"�� J 4�54?L5 , @<; =CE 0 5�� =S; H EB=P8/5 . & M 5IE:5 ,/* 8 576m@ � ce\ tT! 6aN� ���Ra \ %6T&%

GKJN, ; J A 5�5 . 5 A 5IE�83; \ V�$�$ $4V \ % Y B " ��2'?B;/��c � H � L�c � HQ\ T L a �����Qa � Hg\ % T % L�c \ � H T L a�� ���>a \ % � H T % L

42:=S032Q0 J A F . 5 8/5<;�8/2:5+F ,/J�JNG "���2:5 , 5 GKJL, 5N' � H T L V $�$ $4V � H T % L ) c # "

��2:=S; *N. ; J ; 2 J 4 ;�8/2:5 ?B; 5 G ? . , 5<; ? . 88 2 * 8= G Z =>; *�H E:=C8 5 . & M 5<E:5 ,3* 8/5<6 A J 61? . 5 J ?N5 ,*�, =CE M�B @ * E 6�� =>;

* E & ;@? %BA J 61? . 5 JNG Z @ 8 2:5<ER8 2:5 G * 0 8 JN, A J 61? . 5 Z � � =S; H E:=C8 5 . & M 5<E:5 ,3* 8/5<6 "���2:5 * F:F ,/J F , = * 8 5+; ? , � 5<0 8/=R?L5

2 J A J A JN, F:2:=S; A =S; � H T L�c T�a � "

T 5+4= .>. E J 4 5�A1F * EB6 * G * A = . = *N, * EB6!?L5 , & ?B;@5 G ? . 8/2:5 JL, 5 A 8 J A J 61? . 5<;�@:42 * 8 J-G 8 5<EQ0 *-.>. 5<6 8 2:5 HB, ;@8

J

Page 9: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

=S; J A JL, FB2:=>; A 8 2B5 JN, 5 A "

���b�����"$ 1sP � � 5 8�� * E 6 # % 5 A J 61? . 5<; J ?L5 ,�* , =CE MIB!* EB6 . 5 8 � % 5 * E & 2 J A J A JN, F:2:=S; A

GK,/J A � 8 J # " ��2:5IE � � � 0\ � =>; =>; J A JL, F:2B=>0 8 J$� H� L " *�-658554 2 ��2B5 J % ?'= J ?B; A * F:F:=>E M @��"@�42:5 , 5

�"H � a � 0\ � L�c � H � L * 0 8/? *-.>. & 8 ? , EB; J ?:88 J % 5�8 2:5 0 JN,/, 570 8 J E:5�" ��2:5 G * 0 88 2 * 8�� =>;4�5 .>. 615 H E:5<6 * EB6

8 2 * 8�=C8�=S; * EQ=S; J A JN, F:2:=S; A * 8 . 5 * ;@8 ?:FR8 J 8 2:5 J F 5 ,/* 8 = J E J-G a =>; F , J ?L5<6O=CEQ54A * 0 8 . & 8 2:5 ; * A 5+4 * & * ;=>;

6 J E:5 GKJN,�MN,/J ?:F ; ; J =P8+4= .C. % 5 J A =P8 8 576 H � *-.>. = * E V<WLW L�" � J ; 2 J 4 8 2 * 8 �"HQ\ F Lbc&\ ��H F Lt@ E J 8/=>0I5 8/2 * 8 GKJL,

* E & � a � 0\ � Y � � � 0\ � @ 4�5 2 * ?N5 8 2 * 8 ��Hg\`H � a � 0\ � L L,c �"HQ\ � a � 0\ � LOc � Hg\ � L"c \ � H � L,c

\`H���H � a � �\ � L L "��2:=S;0 J A F . 5I8 5<;�8/2:5+F ,/J'J-G "

� ED8 2B5+E:5�A'8�; 5<0 8/= J EO4�5�4= .>. 54A * A =CE:5+; J A 5�F ,/J F 5 , 8 =>5<; J-G�H E:=C8 5 . & M 5IEB5 ,3* 8 576 A J 61? . 5<;�"

Page 10: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

*,+.-0/ �82 ) 9 � � 587:9 ������5 4 � = 3 =g/�� ��� � � 3 � -6+`/��i7 $ 5f7 9 ����

� E 8 2:=S; ; 5<0 8 = J E 4�5�4= .C. F ,/J ?N5�8 2B5�= A F JL, 8 * E�8 8 2:5 JN, 5 A 8/2 * 8�5�?N5 , & ; ? %:A J 61? . 5 J-G * H EB=P8/5 . & M 5<E:5 ,3* 8/5<6

B A J 61? . 5 =>; H EB=P8/5 . & M 5<E:5 ,3* 8/5<6m@ ; JR.CJ E MD* ; B =>; * F , =CE 0 =>F *-. =S615 *N. 6 J A * =>E " T 5 4= .C. *N. ; J 54A1F .CJL, 5 8 2B5

E * 8 ? , 5 J-G *�M 5<E:5 ,3* 8/=CE M ;@5I8 GKJN, 54A * A F . 5M@N42:5I8 2:5 , =C8=S; . =CE:5 *-,/. & 615IF 5IE 615IE�8�" T 2 * 8 GKJN.>.>J 4 ;�=>; * ;@?�� 0I=C5<E�8

0 J EB61=C8 = J E GKJN, * A J 61? . 5�8 J % 5 H E:=C8 5 . & M 5IE:5 ,/* 8 576 "

���b�����"$ �8PQ1 � 5 8 Z�% 5 * A J 61? . 5 * E 6 . 5I8 � % 5 * ; ? %:A J 61? . 5 JNG Z "� G % J 8/2 � * EB6 Z � �

*-, 5 H E:=C8 5 . & M 5<E:5 ,3* 8/5<6 8 2:5<E Z =>; H E:=P8/5 . & M 5IE:5 ,/* 8 576 " *�-6585 4X2 $'=>EB0I5 � * EB6 Z � � *-, 5 H E:=C8 5 . & M 5<E:5 ,3* 8/5<6

A J 61? . 5<;�@ 8/2:5 & % J 8 2 2 * ?L5 M 5<E:5 ,3* 8/=CE M ; 5 83;4@ ; * & . W� V�$�$ $4VuW ; 0 * EB6 . T! a � V�$�$ $4VXT��,a � 0 " T 5 4 * E�8+8 J

; 2 J 4 8 2 * 8Q. T V�$ $�$�V T � VXW V�$ $�$�VuW ; 0 =>; * M 5<E:5 ,3* 8/=CE M ;@5I8 GKJL, Z "� . 5 *-,/. & @-' T V $�$�$�V T � VXW V�$ $�$�VuW ; ) Z " � * $L5

* E &_T Y Z @ 8 2:5<E % 570 * ?B;@5 Z � � =S; H E:=P8/5 . & M 5IE:5 ,/* 8 576m@ 8/2:5 , 5 5�A1=S;�8 , =>E M 5 . 5 A 5IE�83;b\ V�$ $�$�V \ � ;@? 032 8 2 * 8

T a ��ce\ H T& ia ��L aI� ��� a \ � H T��qa �IL�c HQ\ 4T& `a���� �>a \ ��T�� L a � "� . 5 *-,/. & T�Y!T a �@�; J 8 2:5 , 5�54A'=S;@8/;�; J A 5

W"Y � ; ?B032 8 2 * 8 T ce\ 4T& qa�� ��� a \ ��T��sa�W " � 2 J 4�54?L5 , @7=S; H EB=P8/5 . & M 5<E:5 ,3* 8/5<6 *-. ; J @N; J W c ^* 4W� `a�� ��� a�^ ; W ;

GKJN, ; J A 5 ^* V $�$�$�VX^ ; Y B " ��2B5IE T ch\ �T& �a�� ���0a \ �0T��(a]^* 4W� da������ a]^ ; W ; Y ' T! 6V $�$�$4V T��KVXW� 6V�$ $�$�VuW ; ) "

� ?B;@8 * ;�6 J EB5=>E . =>E:5 *-,�*N.CM 5 % ,3* 4�5�0 * E 61=S; 0I?B;/; . =CE:5 *-,/. & 615<F 5<EB615IE�8 * EB6 . =>E:5 *N, . & =CEB6:5IF 5IEB615<E�8 ; ? % ; 5 8/;

J-G A J 61? . 5<;�"

#!�"���4���� ���� �8PQ1 T 5O0 *-.>.*DH E:=C8 5O;@? % ;@5I8 J-G * A J 61? . 5 � @ ; ?B032 * ; . T V $�$�$4V T %M0 @ . =CE:5 *-,/. &

615<F 5<EB615IE�8�= G 8 2:5 , 5 5�A1=>;@8 , =>E M 5 . 5 A 5<E�8/;(\ V $�$ $4V \ % @'E J 8 *-.>. �<5 ,/J @';@? 032 8 2 * 8(\ T a���� �Xa \ % T % ceNq@'42:5 , 5

N =>;�8 2B5 =S615IE�8/=P8 & 5 . 5 A 5<E�8 JNG Z " T 5+0 *-.>.�* ; ? % ; 5 8 J-G�* A J 61? . 5 . =CEB5 *-,/. & =>EB615<F 5<EB615<EL8= G =C8�=S;�E J 8 . =CE:5 *-,/. &

615<F 5<EB615IE�8 " � ; =>E . =>E:5 *-,*N.CM 5 % ,3* 4�5 0 *-.>. * ; ? % ; 5 8 JNG�* A J 61? . 5�� * % * ; =S; GKJN, � = G 8 2:5�; ? % ; 5 8 M 5IEB5 ,3* 8 57;

Z * E 6 = G 8/2:5 ; ? % ; 5 8 =S; . =>E:5 *-,/. & =>EB615<F 5<EB615IE�8 "

� 5 .>J 4 *N, 5�; J A 5 , 57;@? . 8/; , 5 M�*-, 61=>E M % * ; 5<;�"

���b�����"$ �8P � � M 5<E:5 ,3* 8/=CE M ;@5I8 . T V�$ $�$�V T %M0�GKJN,�* E B A J 61? . 5 � =>; . =CE:5 *-,/. & =>EB615<F 5<EB615<EL8 = G

* E 6 J E . & = G 54?L5 , & 5 . 5 A 5<E�8�=>E � 0 * E % 5 , 5IF , 5<; 5IE�8/5<6 * ; * ?:E:= ��?:5 . =CEB5 *-, 0 J A % =>E * 8/= J E J-G 8 2B5�5 . 5 A 5<EL83; =>E

8 2B5 M 5<E:5 ,3* 8/=CE M ; 5 8�" *�-658554X2 � = , ;�8 . 5 8 ? ; * ;/;@? A 5�8/2 * 8�8/2:5 M 5IE:5 ,/* 8 =>E M ; 5 8 =S; . =CEB5 *-,/. & =>EB615IF 5IE 615IE�8�"���2:5<E

* ;/; ? A 5 8 2 * 8 GKJN, ; J A 5 TY Z VXT ch\ 4T& Ka(� ���ga \ %6T&% * EB6 T c ^* tT! <aA� ���ga�^ %6T&% 42B5 , 5�\ <V�$ $�$�V \ %.VX^* V $�$�$4VX^ %

Page 11: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

*-, 5�5 . 5 A 5IE�8/; JNG B " ��2:5IE!N�c TQ@ T c HQ\ D@S^* L T& a � ���4a HQ\ %�@S^ % L T&% @ % ?18 ; =CE 0 5 J ? ,�M 5IE:5 ,/* 8 =>E M ; 5 8�=>;

. =CE:5 *-,/. & =>EB615IF 5IE 615IE�8�@ \<; @]^ ;dc N ; J \ ;�c ^ ; GKJL, *N.C. = c V V $�$�$4V?> "�� J F ,/J ?N5�8/2:5 0 J Ei?N5 , ; 5 * ;/;@? A 5 8 2 * 8

. T V�$�$ $�V T %M0 =S; . =CEB5 *-,/. & 615<F 5<EB615IE�8 "D��2:5<E 8 2:5 , 5 54A1=>;@8/;+5 . 5 A 5IE�8/; J-G 8 2:5 , =>E M \ V�$ $�$�V \ % @ E J 8 *-.>. �I5 , J @

; ?B032 8/2 * 8O\ T a���� � a \ % T % chN T a N T " a���� �0a N T % " ��2'?B;ON 2 * ;�8�4 J 61=S;�8/=CEB0 8 , 5<F , 5<; 5IE�8 * 8 = J EB;�"

��5�A'8�4�5�2 * ?N5 * ER=>EL8/5 , 5<;@8 =>E M . 5 A A * 8 2 * 8 F , J ?L5<; A * E & ; = A F . 5 , 5<; ? . 8/; GKJL,,B A J 61? . 5<;42B5 , 5 B =>;

* ED=>EL8/5 ML,/*N. 6 J A * =>E "

���b�����"$ �8P�� � G(B =S; * E =>E�8 5 ML,/*N. 6 J A * =>E * E 6 � =S; * E B A J 61? . 5 4=P8/2 *OM 5<E:5 ,3* 8/=CE M ;@5I8

. T! 6V�$�$ $�V T&% 0 @ * EB6O= G ��c . W� V $�$�$4VXW�� 0 =S; * E & ;@? % ;@5I8 JNG � 42:5 , 5 ��� > @'8 2:5<E ��=>; * . =CE:5 *-,/. & 615<F 5<EB615<EL8

; 5 8�" *�-6585 4X2 � J 8 =S0 5R8 2 * 8 W ; Y � GKJN, *N.C.�= c V V�$�$ $�V ��" ��2'?B; 4�5Q2 * ?N5R5 . 5 A 5<E�8/; ����� Y B @�42:5 , 5

F c V V�$ $�$�V � * EB6 � c V V $�$�$4V?> ;@? 032 8/2 * 8

W� c � ? 4T& a���� � a � % T&%W " c � " 4T& a���� � a � "?% T&%" " "W � c�� �� T a�� ���0a � � % T %

� J 8 =S0 5M@ 8 2 * 8�4�5 4 * E�8�8 J ; 2 J 498 2 * 8 8/2:5 , 5 54A'=S;@8 , =CE M 5 . 5 A 5IE�83; � 8 JA��� @�E J 8 *-.>. �<5 ,/J @�;@?B032 8 2 * 8

� W a���� � a �� W � c N.@142:=S032O4 J ? . 6DF ,/J ?N5�8 2 * 8B. W V�$�$ $�VXW � 0 =S; . =CEB5 *-,/. & 615IF 5IE 615IE�8�" � J 4 0 J EB; =>6:5 , 8 2B5

GKJN.>.>J 4=CE M ; & ;�8/5 A JNG 5���? * 8 = J EB;�"

�� � a�� ���0a �� � � ceN" " "�� � % a���� � a �� � � % c N

� G 8/2:=>;�; & ;�8/5 A J-G 5 �L? * 8 = J E ; 2 * ; * E J E! 8 , =R?'= *-. ; JL. ?18/= J EQ=CE B @ 8/2:5IEQ8/2 J ; 5�? *N. ?:57; JNG �� 8 J �� *N. ; J

A * $L5 � �W� aL��� �Ka � �QW� c�N " � 5I8 D % 5 8/2:5 H 5 . 6 JNG ��? J 8 =>5IE�8/; GKJN,bB H � *-.>. = * E � � ML�" ��2B5IE"8 2:5 , 5 *-, 5

5 . 5 A 5<EL83; � V $�$�$�V � ��Y D E J 8 *N.C. �I5 ,/J 8/2 * 8 ; JN. ?N5 8 2:5 ; & ;�8/5 A ; =CEB0I5 ��� >R% 570 * ?B;@5 4�5 0 * E , 5<61? 0 5+8/2:=>;

; & ;@8 5 A J ?L5 ,�D @ JL, =CE J 8/2:5 , 4 JL, 6:;�8/2:5+E'? A % 5 ,�J-G 0 JN. ? A EB;=S; ML, 5 * 8 5 , 8 2 * ED8 2:5 E'? A % 5 ,J-G�,/J 4 ;�=>ER8 2B5

0 JL, , 5<; F J EB61=>E M+* ? M A 5IE�8 576 A * 8 , =RA " � ?18�4�5�0 * E A ? . 8/=CF . & 8 2 ,/J ? M 2 %'& 8/2:5�F ,/J 61?B0 8 J-G 8/2:5�615<E J A =>E * 8 JN, ;

J-G 8 2:5 � ; � ; H J ?N5 , V L 8 J�M 5I8 ; JN. ?18/= J EB; 8/2 * 8 *-, 5=>E B " ��2:5 , 5 GKJL, 5 8 2:5; 5 8 . W V $�$ $4VXW � 0 =S; . =>E:5 *N, . & 615IF 5IEB6:5IE�8�"

��2:5 , 5 *-, 5 8�4 J = A F JL, 8 * E�8�= A A 5<61= * 8/5+0 JN,/JN.>.S*-, =C57; 8 J 8 2:=S; . 5 A A * "���2:5 HB, ;�8�=>;�8/2 * 8 GKJN,* E & =>EL8/5 ML,/*N.

6 J A * =CE B @:5�?N5 , & % * ;@=S; J-G�* E & B A J 61? . 5+2 * ;�8/2:5 ; * A 5+;@= �I5M@B;@=>EB0I5+= G�J EB5 % * ; =>;�4�5 , 5 ; A *N.C. 5 , @'8 2B5IER8 2B5

Page 12: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

.S*-,/M 5 , % * ; =>; 4 J ? . 6 % 5 . =>E:5 *N, . & 6:5IF 5IEB615<E�8�"���2:5 ; 5<0 J EB6 =>; 8 2 * 8 * % * ; =S;�=>; * ; A *-.>. 5<;@8 F J ; ; = % . 5 M 5IE:5 ,/* 8 =>E M

; 5 8 GKJN,�* A J 61? . 5�"�� 8 ; 2 J ? . 6 % 5�E J 8 5<6 8 2 * 8 * . =>E:5 *N, . & =CEB6:5IF 5IEB615<E�8;@5I8 JNG A * A1= A *N. ;@= �I5�=S;�E J 8�E:5<0I5<;/; *N, = . &

* M 5IE:5 ,/* 8 =>E M ;@5I8�"� JL, 5�A * A F . 5M@B=>ER8 2:5 H E:=P8/5 . & M 5IEB5 ,3* 8 576I� A J 61? . 5�H * EB6 *-. ; J � % 5 . = * E MN,/J ?:FsL��b@&. � 0

=S; * . =>E:5 *-,/. & =>EB615<F 5<EB615<EL8D;@5I8 J-G A * A1= A *-. ; = �I5Q=>E � % ?:8 =P8 6 J 57; E J 8 M 5IEB5 ,3* 8 5�� " � . ; J * A =CEB= A *-.

M 5<E:5 ,3* 8/=CE M ;@5I8�=>;�E J 8 *-. 4 * & ; * % * ;@=S;�@ GKJN, 5�A * A F . 5K@ . H V V N L V HQN V V L 0 =S; * A =>E:= A *-. M 5IE:5 ,/* 8 =>E M ; 5 8 GKJL, 8 2B5

� % 5 . = * E MN,/J ?:F � P ���t@ *-. 8 2 J ? M 2O=C8 =>;�E J 8 . =CE:5 *-,/. & =>EB615<F 5<EB615<EL8�;@=>EB0 5 sHQN V V L c�HQN V NMLdchN "

� 5 8N� % 5 *RH E:=P8/5 . & M 5IE:5 ,/* 8 576 B A J 61? . 5 4=C8 2 * M 5IEB5 ,3* 8 =>E M ; 5 8 JNG ;@= �I5 > "U��2:5 , 5 =S; * E * 8 ? ,/*N.

; ? , � 570 8/=R?L5 2 J A J A JN, F:2:=S; A GK, J A 8 2B5 B A J 6:? . 5 B % c BLP BLP � ��� PeB J E�8 J � " $'?:F:F J ; 5 8 2 * 8�� � ;

M 5<E:5 ,3* 8/=CE M ; 5 8 =S;Q. T& 6V�$ $�$�VXT#% 0 "��2:5 G ?BEB0 8/= J EQ4�5+4 * E�8�8 * $N5<;�Hg\ <V�$ $�$�V \ % L�8 J \ 4T! da������6a \ %<T&% "�� J 8/=>0I5

8 2 * 8�= G�J ? , M 5IE:5 ,/* 8 =>E M ; 5 8 =S; . =CE:5 *-,/. & =CE 615IF 5IEB6:5IE�8�@ * EB6O8/2�? ; * % * ;@=S;�@18 2:=S; A * F:F:=>E M =S; J E:5� 8 J J E:5 * EB6

8 2'?B; * ED=>; J A JN, F:2:=S; A "

T 5O4= .>. E J 4 F , J ?L5 8 2 * 8 GKJL, * F , =>EB0I=CF *N. =S615 *-. 6 J A * =>E B H � *N.C. = * E � � � L�@�54?L5 , & ; ? %:A J 61? . 5 JNG�*

H E:=C8 5 . & M 5IE:5 ,/* 8 5<6 B A J 61? . 5=S; H EB=P8/5 . & M 5IE:5 ,/* 8 576 "�� = , ;@8�@�2 J 4�54?N5 , @N4�5 4= .>. F ,/J ?N5�8/2:5 GKJN.>.>J 4=CE M+. 5 A A * @

8 2 * 8 54?L5 , & ;@? %:A J 6:? . 5 JNG B % =>; H E:=P8/5 . & M 5IE:5 ,/* 8 5<6 %'& * ;@5I8 J-G ;@= �I5 > "

���b�����"$ �mP�C T 2:5IE B =S; * F , =CE 0 =>F *-. =S615 *N. 6 J A * =CE8@�54?L5 , & ;@? %BA J 61? . 5 JNG 8 2B5 B A J 61? . 5

B % =S; H E:=C8 5 . & M 5IE:5 ,/* 8 576 * EB6 2 * ; *UM 5<E:5 ,3* 8/=CE M ; 5 8 J-G ; =��<5 > " *�-6585 4X2 T 5 4= .>. ?B; 5 =>EB61? 0 8 = J E J E > "

� JL, 8 2B5 % * ; 5 0 * ;@5 JNG > c V @�4�5 *-, 5 .CJ'J $'=>E M * 8 8/2:5U; ? %:A J 61? . 5<; JNG B J ?N5 ,B =C8/; 5 .CG @42:=S032 * ; 4�5

*-.>, 5 * 6 & E J 8/5<6 *-, 5 8/2:5 =>6:5 *-. ; J-G B " $'=>EB0I5 B =>; * F , =CE 0 =>F *-. =S615 *N. 6 J A * =CE"54?N5 , & =S615 *-. * E 6 8/2�? ;�54?N5 , &

; ? %:A J 61? . 5D=S; J-G 8 2:5 GKJL, A 'Q\ ) @�42:5 , 5 \U=S; � ?B;@8 * E 5 . 5 A 5IE�8 J-G�B @ * EB6�8 2'?B; 5�?N5 , & ; ? %:A J 61? . 5 JNG�B

2 * ; * H E:=P8/5 M 5IEB5 ,3* 8 =>E M ;@5I8�" � J 4 . 5I8+?B; * ;/;@? A 5 8/2 * 8 GKJL, ; J A 5 =>EL8/5 M 5 , > @ 5�?N5 , & ; ? %:A J 61? . 5 J-G B % =>;

H E:=C8 5 . & M 5IE:5 ,/* 8 5<6 " � J EB; =>6:5 , 8/2:5 2 J A J A JN, F:2:=S; A �"J E�8 J 8 2:5 HB, ;@8 > ; ? A A * EB6:; GK, J A B %�� 8 J�B % @ =>E

J 8/2:5 , 4 JL, 6B;�8 2:5 A * F:F:=>E M � H � <V�$ $�$�V � %`V � %�� L c H � <V�$ $�$�V � % L�"���2:5�$N5 , E:5 . JNG 8/2:=S;�2 J A J A JN, F:2:=S; A

=S; 0 . 5 *N, . & 8 2B5 ;@? % MN,/J ?:F . H �� V�$ $�$�V �I% V �I%�� L Y B %�� � � ;,c N V = c V V�$�$ $4VK> 0 42:=>032 =S;+=S; J A JL, FB2:=>0 8 J

B @ * EB6 8 2'?B; H E:=C8 5 . & M 5<E:5 ,3* 8/5<6eHK4=C8 2 * ; =CE ML. 5 M 5<E:5 ,3* 8 JN, L�" � J 4 . 5 8 � % 5 * E & ; ? %:A J 61? . 5 JNG,B %�� "

��2:5<E � H�� L�=>; * ; ? %:A J 61? . 5 J-G�B % @ * EB6U8 2'?B; %'& 8/2:5 =>EB61?B0 8 = J E 2 & F J 8/2:5<; =S; H E:=C8 5 . & M 5IE:5 ,/* 8 5<6"4=C8 2 *

M 5<E:5 ,3* 8/=CE M ; 5 8 JNG ; = �I5 > "�� 5 8 � % 5�8 2B5 $N5 , E:5 . J-G 8 2 * 8 A * F:FB=CE M GK,/J A � 8 J � H�� L�@ ; J � =>;=S; J A JN, F:2:=S0

W

Page 13: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

8 J * ; ? %:A J 61? . 5 JNG�B * EB6 =>; M 5IE:5 ,/* 8 5<6 %'& * ;@=>E MN. 5D5 . 5 A 5IE�8�@�; =CEB0I5 B =>; * F , =>EB0 =>F *-. =S615 *-. 6 J A * =CE "

� & 8/2:5 HB, ;@8�=S; J A JN, F:2:=S; A 8 2:5 JN, 5 A GKJN, A J 61? . 57;4@ � H�� L =>; =S; J A JL, FB2:=>0�8 J � � � "���2'?B;�� � � =>; H E:=C8 5 . &

M 5<E:5 ,3* 8/5<6"4=P8/2 *DM 5IEB5 ,3* 8 =>E M ; 5 8 J-G ; =��<5 > " ��2:5 , 5 GKJN, 5 %'& 8 2:5 JN, 5 A � " V @�; =CEB0I5 % J 8 2�� * E 6 � � � *-, 5

H E:=C8 5 . & M 5IEB5 ,3* 8 576m@ � =S; * H E:=P8/5 . & M 5IE:5 ,/* 8 5<6 B A J 61? . 5 4=C8 2 * M 5IEB5 ,3* 8 =>E M ;@5I8 JNG ; =��<5 >�a V @ H J-G 0 J ? , ; 5

=C8 A = M 2�8 2 * ?L5 * ; A *-.>. 5 ,M 5IEB5 ,3* 8 =>E M ;@5I8uL "

��2:=S; , 5<; ? . 8 * E 6 8/2:5 E:5�A'8 J E:5 0 * EU5 * ; = . & * E 6 J % ?'= J ?B; . &D% 5 5�A'8 5<EB61576 8 J 8/2:5 0 * ;@5 42:5IE 4�5 2 * ?L5

*D, =>E M�B 4=C8 2 8 2B5 F , J F 5 , 8 & 8/2 * 8 5�?N5 , & =S615 *N. =>; H E:=P8/5 . & M 5IE:5 ,/* 8 5<6 " � J 4�5�?N5 , @ ; =CEB0I5 42 * 8+4�5 4= .>. % 5

615 *-. =CE M 4=P8/2U=>E 8 2B5 E:54A�8�;@570 8/= J E *-, 5 � ? 0 . =>615 * EU6 J A * =>EB; 42B=>032 *N, 5 F , =>EB0I=CF *N. =S615 *-. 6 J A * =>EB;�@ .>J'J $'=>E M

* 8�8 2:=S; , 5<;@8 , =S0 8 576Q0 * ;@5 =S;�; ? � 0 =>5IE�8 " � EQ8 2B5 M 5<E:5 ,3*-. 0 * ; 5 42:5 , 5 54?N5 , & =>6:5 *-. J-GdB =S; H E:=C8 5 . & M 5IE:5 ,/* 8 5<68@

; J A 5 ; ? %:A J 61? . 5 JNG(B % A = M 2�8�2 * ?L5 * A =>E:= A ? A M 5IEB5 ,3* 8 =>E M ; 5 8 J-G ; = �I5 .>*N, M 5 , 8/2 * E > " � ? ;�8 * ; * ; =>6:5

E J 8 5K@�8 2B=>; F , J F 5 , 8 & 8 2 * 8 54?L5 , & =S615 *N. =S; H E:=C8 5 . & M 5<E:5 ,3* 8/5<6 =S;+5���?:=p? *-. 5IE�8 8 J 8 2:5 0 J EB6:=P8/= J E"8/2 * 8 54?N5 , &

;@8 , =>0 8 . & =CEB0 , 5 * ; =CE M 032 * =CE JNG =S615 *N. ; =S; H E:=P8/5R=CE . 5<E M 8/2[H � *N.C. = * E � �MLt@ JN, =>E J 8 2:5 , 4 JN, 6:; 8 2 * 8 B =S; *

� J 5I8 2:5 , = * EU6 J A * =>E " T 5 4= .C. E J 4 F ,/J ?N5+8/2:5 A * =CE , 57;@? . 8 J-G 8/2:=>;�; 5<0 8 = J E8@ 8 2 * 8�54?N5 , & ; ? %:A J 61? . 5 JNG*

H E:=C8 5 . & M 5<E:5 ,3* 8/5<6 B A J 6:? . 5+=S; H E:=C8 5 . & M 5<E:5 ,3* 8/5<6m@:; J .CJ E M * ; B =S; * F , =CEB0I=CF *-. =S615 *N. 6 J A * =>E "

���b�����"$ �mP � � 5 8 B % 5 * F , =CEB0I=CF *-. =>6:5 *-. 6 J A * =>E8@�8 2B5IE 54?N5 , & ; ? %:A J 61? . 5 JNG�*UH E:=C8 5 . &

B A J 61? . 5 =S; H E:=C8 5 . & M 5IE:5 ,/* 8 5<6 " *�-658554 2 � 5I8�� % 5 * H E:=P8/5 . & M 5IEB5 ,3* 8 576 B A J 6:? . 5 * EB6 . 5 8�# % 5 * E &

; ? %:A J 61? . 5 JNG � " � J 4L� 2 * ; * H E:=P8/5 M 5IEB5 ,3* 8 =>E M ;@5I8 ; * & +Uc . T V T " V�$�$ $4VXT %�0 "� J EB; =>6:5 , 8 2:5+E * 8 ? ,/*N.

; ? , � 570 8/=R?L5 2 J A J A JL, FB2:=>; A @ � ( B % � � 615 H E:576 %'& � HQ\ <V�$�$ $4V \ % Ldch\ 4T! ma \ "4T#"fa � ��� a \ %6T&% " T 5 $'E J 4

8 2 * 8 � H# L c . F Y B % � � H F L Y # 0 =S; * ;@? %:A J 6:? . 5 J-GOB % "O��2:5 , 5 GKJN, 5 %�& 8 2:5 JN, 5 A �!" �q@ � H#SL�=>;

H E:=C8 5 . & M 5IEB5 ,3* 8 576 " � J 4 ; =>EB0 5 � =S; * ; ? , � 5<0 8/=R?L5 2 J A J A JN, F:2:=S; A J E�8 J # 42:5<E =C8/;6 J A * =>E =S; , 57;�8 , =S0 8/5<6

8 J �� H#SLt@ %'& V " J.@�# =S; H E:=C8 5 . & M 5<E:5 ,3* 8/5<6 "

T 5 E J 4 2 * ?N5 * E:=S0 5 % =S0 J EB61=C8 = J E *N. ;�8 * 8/5 A 5IE�8 * % J ?:8 H E:=P8/5 . & M 5IE:5 ,/* 8 576 A J 61? . 5<; * EB6U8 2:5<= , ; ? %

A J 61? . 5<;�"

���b�����"$ �mP�! � GqB =>; * F , =>EB0 =>F *-. =S615 *N. 6 J A * =CEf@*� =S; * E B A J 61? . 5 * EB6 # =S; * E & ; ? %:A J 61? . 5

J-G � @78/2:5IE�� =S; H E:=C8 5 . & M 5<E:5 ,3* 8/5<6�= GB* EB6 J E . & = G # =S; H EB=P8/5 . & M 5IE:5 ,/* 8 576 * EB62� � # =>; H E:=C8 5 . & M 5IE:5 ,/* 8 5<6 "

*�-658554 2 ��2:=S; J-G 0 J ? , ;@5 GKJN.>.CJ 4 ;�= A A 5<61= * 8/5 . & GK,/J A 8/2:5 JL, 5 A ; V " J.@ �!" V * EB6 �!" J!"

V N

Page 14: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

*,+.-0/A�f2 �%� +.;K/�5s- = � +`/0=Q5 3 554 � = 3 =g/ � ��� � � 3 � -6+`/��i7 $ 5f7 9 ����

T 5 4= .C. E J 4 F ,/J ?N5 J ? , A * � JN, , 57;@? . 8�42B=>032Q=S;8/2 * 8�54?N5 , & H E:=C8 5 . & M 5<E:5 ,3* 8/5<6 B A J 61? . 5K@ 42:5 , 5 B

=S; * � ?B0 . =S615 * EQ� J A * =>E8@:=S;=>; J A JL, F:2B=>0 8 J 8/2:5 61= , 5<0 8 F , J 61?B0 8 J-G B A J 61? . 5<; JNG 8/2:5 GKJL, A B � ' T ) @ 42:5 , 5

T =S; * ER5 . 5 A 5IE�8 JNG B " � ?18 HB, ;@8�@ 4�5+EB5I5<6 * G 5<4 A JL, 5�8/2:5 JL, 5 A ;�"�� 5 8�? ; 0 J EB;@=S615 , * E & 2 J A J A JN, F:2:=S; A

� ( B�� � B�� " T 5+4= .>. ; 2 J 4 8/2 * 8 � =>; * A * 8 , =RA 8 ,/* EB; GKJL, A * 8 = J E "

���b�����"$ �fPQ1 � 5 8 � % 5 * E ��� � A * 8 , = A J ?N5 ,�*+, =>E M�B "���2:5IE 8 2:5 , 5 =>; * 2 J A J A JN, F:2:=S; A

� ( B � � B � ;@?B032 8 2 * 8 � H F L�c � F " *�-658554 2 � J EB; =>615 , 8 2:5I��� � A * 8 , = A � @ * EB6 8 2:5 A * FBF:=CE M

� ( B � � B � @�42:5 , 5 � H F L,c�� F "+� 5I8 F * EB6 � % 5 =>E B � * EB6 . 5I8"\ % 5 =>E B @ ; J 4�5 2 * ?N5 � H F a � L�c

� H F a � L c:� F a � � c � H F L a � H � L * E 6 � Hg\ F L,c:� Hg\ F LOc H� \<L F c \*� F c \ � H F L�"+��2'?B; � =>; *

2 J A J A JN, F:2:=S; A " ��2:5 , 5 GKJL, 5 GKJN, 54?L5 , & ��� � A * 8 , = A � 4=P8/2O5 . 5 A 5<E�8/;�=CE B 8 2B5 , 5 =S; * 2 J A J A JN, F:2:=S; A

� ( B � � B � ;@? 032 8/2 * 8 � H F L�cL� F "

� J 4 4�5�4= .>. F ,/J ?N5 8 2:5 0 J Ei?N5 , ; 5 JNG 8/2:5 * % J ?N5�;�8 * 8/5 A 5IE�8�"

���b�����"$ �8P�� � 5I8 B % 5 * 0 J A A ?:8 * 8/=R?L5 , =>E M " ��2:5<E 54?N5 , & 2 J A J A JN, F:2:=S; A % 5I8�4�5<5IE 8 2B5

B A J 61? . 5<; B � * EB6 B � =>; * A * 8 , = AO8 ,3* EB; GKJN, A * 8 = J E " *�-6585 4X2 � 5 8 � ( B � � B � % 5 * 2 J A J A JL, F:2B=>; A "

T =C8 2 J ?18 .>J ;/; JNG�M 5IE:5 ,/*N. =C8 & 4�5�0 * E 8 2B=CE $ JNG 8/2:5�5 . 5 A 5IE�8/; J-G8B � * ;�0 JL. ? A E�?L5<0 8 JN, ;�" � 5I8?B;�615 H EB5� ; * ;

8 2B5 = � 8/2R0 JN. ? A E J ED8 2:5 � � � =>6:5IE�8 =C8 & A * 8 , =RA "� J 8 =S0 5�8/2 * 8 � =>;�0 J A F . 5 8/5 . & 615I8 5 , A =>E:5<6 %'& 2 J 4 8 2B5

61=� 5 , 5<E�8 ; � ; *N, 5 A * F:F 5<6m@1; =CEB0I5 * E & 2 J A J A JN, F:2:=S; A =>;�0 J A F . 5I8 5 . & 615 8/5 , A =CE:576 %'& =P83; % 5I2 * ?'= JN, J E 8 2B5

M 5<E:5 ,3* 8/=CE M ; 5 8�" � G 4�5�615 H E:5�8 2:5��� � A * 8 , =RA � ; J 8 2 * 8 = � 8/2O0 JL. ? A ED=S; � ? ;�8 � H ; Lt@'8 2B5IE � H F L�cL� F "

��2'?B;�54?L5 , & 2 J A J A JN, F:2:=S; A � ( B � � B � =S; * A * 8 , = A 8 ,3* EB; GKJN, A * 8 = J E "

T 5 2 * ?L5 � ? ;�857;�8 * % . =S;@2:576 * 0 JN,/, 57;@F J E 615IEB0I5 % 5I8�4�5<5IE A * 8 , =S0 5<; * EB6 2 J A J A JL, FB2:=>; A ; 42:=S032 4= .>. % 5

?'=P8 *-. =CE J ? , F ,/J�JNG ;@2 JL, 8 . & " � ?18 HB, ;�8 8/2:5 , 5 *N, 5 * 0 J ?:F . 5 JNG =CE�8 5 , 57;�8/=CE MO* E 6QF ,/J % * % . & 5�A1F 5<0 8 5<6 F J =CE�8/;

42:=S032 0 * E % 5 A * 615 * % J ?18 8 2B5E * 8/? , 5 J-G 8 2B5 A * 8 , = A�� * EB6 =C8/; , 5 .S* 8 576 2 J A J A JN, F:2:=S; A "�� G 8/2:50 JL. ? A EB;

J-G � *-, 5 . =>E:5 *N, . & =>EB615<F 5<EB615IE�80@ 8/2:5IE 8/2:5 2 J A J A JN, F:2:=S; A =S; * E =>E � 570 8 = J E "O��2:=S;+=S;+5 * ; & 8 J ;@5<5 ;@=>EB0I5

J E . & 8 2B5 �I5 , J ?N5<0 8 JL, =>; A * F:F 5<6 8 J �<5 ,/J JN, =CE J 8/2:5 , 4 JL, 6:;�� � 0\iH � L��qc V " � . ; J @ = G 8/2:5 0 JN. ? A EB; J-G �

; F * E B � 8/2:5IE J ? , 2 J A J A JL, FB2:=>; A =>; * ; ? , � 5<0 8/= J E "�� JN, 8 * $N5 * E & F Y B � 8 2:5<E8@ ; =CEB0I5 8/2:5 0 JN. ? A EB; J-G

VNV

Page 15: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

� *-, 5 * ;@F * E:E:=>E M ; 5 8�@ 4�5 0 * ER4 , =P8/5 F c \ ��� da ��� � a \���42B5 , 5b\ 8/2 ,/J ? M 2!\

�*N, 5�5 . 5 A 5<EL83; JNG B

* E 6 � 8/2 ,/J ? M 2 ��*N, 5�8 2B5 0 JN. ? A EB; JNG � "���2'?B; � H HQ\ V�$ $�$�V \

�L Ldc F "

T 5�4= .C. E J 4 F ,/J ?N5�8/2 * 8�= G B =S; * �?B0 . =S615 * E 6 J A * =>E8@ H * EB6 8 2'?B; * F , =>EB0 =>F *-. =S615 *N. 6 J A * =CE * EB6 *N. ; J

* ?:E:= ��?:5 G * 0 8 JN, =�� * 8/= J EO6 J A * =CE_H � *N.C. = * E V<W L Lt@'8/2:5IED54?L5 , & H E:=C8 5 . & M 5IEB5 ,3* 8 576 B A J 61? . 5�=>;�=S; J A JN, F:2:=S0

8 J * E B A J 6:? . 5 J-G 8/2:5 GKJL, A B � ' T ) P ��� � PSB � ' T %*) @�8/2:5 61= , 5<0 8�F ,/J 61?B0 8 J-G 0 & 0 . =S0 B A J 6:? . 57;�" � = , ;@8�4�5

4= .C. F , 57;@5<EL8 * ; A *N.C. . 5 A A * "

� � � � +m2 � G�B =S; * � ? 0 . =>615 * E�6 J A * =>E8@ * EB6�= G \ Y B * EB6 \Q=>; * ?:EB=P80@�8 2:5<E o�Hg\<L�r o�H F L GKJL,

*-.>. E J E! �I5 , J F Y B " ��2:=S; =S; 5 * ; & 8 J F , J ?L5 � ?B;@8 E J 8/5 8 2 * 8 %�& 8/2:5D6:5 H E:=C8 = J E J-G �?B0 . =>6:5 * E 6 J A * =>E

o�Hg\<L�r[o�Hg\ \ L�c�osH V L�r osH V F LOc osH F L " � . ; J =P8�=>; 5 * ; & 8 J ;@5<5 8/2 * 8�= G osH F L,c osH V Lt@ 8/2:5IE F =S; * ?:E:=C8�"

� & 8/2:5 615 H E:=P8/= J E J-G* � ?B0 . =S615 * E � J A * =CE 8 2B5 , 5 5�A1=>;@8 5 . 5 A 5<EL83;6� * EB6S\O=>E B ; ?B032 8 2 * 8 V c � F a \

42:5 , 5�\c�N JL, o�Hg\<L2� o�H F L�" � ?18bo�Hg\<LB� o�H F L =S; * 0 J E�8 ,3* 61=S0 8/= J EU;@=>EB0I5�o�H F L�c o�H V L * EB6 V =S; * ?:E:=C8�"

��2'?B; V c � F a N�c � F * EB6 F =S; * ?:E:=C8�"

���b�����"$ �8P � � G�B =>; * � ?B0 . =S615 * E 6 J A * =>E8@�8 2B5IE 54?N5 , & H EB=P8/5 . & M 5<E:5 ,3* 8/5<6 B A J 61? . 5D=>;

=S; J A JL, FB2:=>0�8 J * E B A J 61? . 5 JNG 8/2:5 GKJN, A B � ' T! )-P � ��� P B � ' T&% ) 42:5 , 5 T& 6V�$ $�$4VXT&% Y B " *�- 5f5 4X2 � 5 8 B

% 5 * � ? 0 . =>615 * E 6 J A * =>E * EB6 . 5 8 � % 5 *�H E:=P8/5 . & M 5IE:5 ,/* 8 576 B A J 6:? . 54=P8/2 *M 5<E:5 ,3* 8/=CE M ; 5 8�. T V $�$�$�V T %�0 "

� J E ;@=S615 , 8/2:5�E * 8/? ,/*N. ;@? , � 570 8 =p?N5�2 J A J A JN, F:2:=S; A @ � ( B % � � 42B5 , 5 � H F- V�$�$ $�V F % L�c F T a������>a F5% T % "

� 5I8 ?B;�0 *N.C. 8 2:5�$N5 , E:5 . JNG 8 2:=S;�2 J A J A JN, F:2:=S; A � " $'=>EB0 5 � =S; * ; ? %:A J 61? . 5 JNG�B % 4�5�$�E J 4 8/2 * 8�=P8�=>;

H E:=C8 5 . & M 5IEB5 ,3* 8 576m@ * EB6 2 * ; * M 5<E:5 ,3* 8/=CE M ; 5 8 J-G ; = �I5 > @1; * & . W� 0V $�$�$�VXW % 0 "� J EB; =>6:5 , 8 2:5�2 J A J A JN, F:2:=S; A

� ( B % � � 42:5 , 5 ��H H F <V�$�$ $4V F % LOc F �W� �aL� ���<a F %<W % " � ?18�8 2:5<E �9=>; � ?B;@8 * A * 8 , =RAD8 ,/* EB; GKJN, A * 8 = J E

�"H � L(c�� � 42B5 , 5�8/2:5 0 JL. ? A E ; J-G 8/2:5 A * 8 , =RA�� *-, 5�8/2:55 . 5 A 5<EL83; J-G 8 2:5 M 5IE:5 ,/* 8 =>E M ;@5I8 . W� V $�$ $�VXW % 0 " $ J

GKJN, 5�?N5 , & H E:=C8 5 . & M 5IEB5 ,3* 8 576 A J 61? . 5�8 2:5 , 5�=>; * > � > A * 8 , = A�� * EB6 =P83;0 JN,/, 5<; F J EB61=>E M 2 J A J A JN, F:2:=S; A

� ; ?B032 8/2 * 8�� =>; =>; J A JN, F:2:=S0�8 JB % � ��H B % L %'& 8 2B5 HB, ;@8=>; J A JL, F:2B=>; A 8 2:5 JN, 5 A "

� J 4 0 J EB; =S615 , 42 * 82 * F:F 5IEB;�= G �9=>; * 61= *-MLJ E *N. A * 8 , =RA 4=C8 2 5<E�8 , =C57; � V�$�$ $4V � %�*-.>J E M 8 2:5�61= *NMNJ E *-. "

��2:5<E �%c '�� ) P '�� " ) P ��� � P '� % ) " � J 8 =S0 5+8/2 * 8�= G ��; c Nq@:8 2B5IE�'�� ; ) c�.0N 0 * EB6R= G '��; ) =S; * ?:E:=C8�@ 8 2:5<E

'��; ) c B " � J 4 .CJ'J $ * 88 2B5���? J 8/=C5<E�8 A J 61? . 5 B % � H '� �)!P � ��� P '�� % ) L�"���2:=S; A J 61? . 5 =S;�=>; J A JN, F:2:=S0�8 J 8 2B5

B A J 61? . 5 B � '� )/P B � '� " ) P � ��� P�� � '� % ) 4=P8/2 8/2:5 J % ?'= J ?B; A * F:F:=>E M " � J 8 =S0 5 8 2 * 8 B � 'QN ) =S; � ?B;@8 B

V �

Page 16: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

* E 6 B � '�� ) =S;=S; J A JN, F:2:=S0�8 J . N 0 42:5<E:54?L5 , � =S; * ?BE:=P8 " T 5�0 * E 5<;/;@5<EL8/= *N.C. & 8/2 ,/J 4 * 4 * & JL,�GKJN,/M 5I8 * % J ?:8

8 2B5�8 2:5 G * 0 8 JL, ; JNG 8/2:5 GKJN, A B � '�� ) @ * ;�8 2B5 & * 6:6OE J 8/2:=CE M 8 J 8 2:5 A J 61? . 5 "

T 5 4= .C. E J 4 .CJ'J $ * 8�42 * 8�2 * F:F 5IEB;�= G ��=>;�E J 8 * 61= *NMNJ E *-. A * 8 , = A " T 5 4= .>. ; 2 J 4 2 J 4 8 J , 5<61?B0I5

� 8 J � � @ * E 6 8 2 * 8 B � � =S; =S; J A JN, F:2:=S0R8 J B � � � @�42:5 , 5$� � =>;O8 2B5U= A *NM 5 J-G 8 2B5"2 J A J A JN, F:2:=S; A

0 JL, , 5<; F J EB61=>E M 8 J 8 2:5 , 5<61? 0 5<6 A * 8 , =RA�� � " T 5Q4= .>.�HB, ;�8 ; 2 J 4 8/2 * 8 8 2B5 GKJN.>.CJ 4=>E MU,/J 4 * EB6 0 JN. ? A E

J F 5 ,3* 8/= J EB; J E �jHK42:=S032 J-G 0 J ? , ; 5 0 J ? . 6 032 * E M 5 8 2B5 2 J A J A JL, FB2:=>; A � L 6 J E J 8 032 * E M 5 8/2:5 A J 61? . 5

B � � ?:FD8 J =S; J A JN, F:2:=S; A " T 5+4= .C. ?B;@5+=S615 * ; J-G 5 . 5 A 5IE�8 *-, & A * 8 , =S0 5<;�8 J F , J ?L5�; J A 5 J-G 8 2B5<; 5 , 5<; ? . 8/;�"

* " L � JN. ? A EU�5IF .>* 0 5 A 5<EL8 ( ��2:=S;�=S; * 6:61=>E MO* ; 0 *-.S*-, A ? . 8 =>F . 5 J-G J E:5 0 JN. ? A EQ8 JD* E J 8 2:5 , "+� . 5 *-,/. &

8 2B=>; 6 J 5<;+E J 8 032 * E M 5 8/2:5 A J 61? . 5 � c ��H B % L�" $'=>EB0I5 � 2 * ; *RM 5IEB5 ,3* 8 =>E M ; 5 8 . W V $�$�$4VXW %M0 * EB6 8 2B5

A * 8 , =RA � 0 J E ;@=S;�83; JNG A 5 , 5 . & 8/2:5O5 . 5 A 5IE�8/; J-G 8/2 * 8 ;@5I8 * ; 0 JN. ? A EB;4@ * 6:61=>E M ; J A 5O;/0 *N.>*N, A ? . 8/=CF . 5 J-G

0 JL. ? A E = 8 J 0 JN. ? A E��Q=>; , 5 *N.C. & � ?B;@8 032 * E M =CE M 8/2:5 M 5<E:5 ,3* 8/=CE M ;@5I8+8 J . W 0V�$ $�$4VuW ; V $�$ $�V \ W ; aeW �KV�$�$ $�VXW�% 0

GKJN, ; J A 5 , =>E M 5 . 5 A 5IE�8,\i@ % ?18 JNG 0 J ? , ; 5�8 2 * 8 , 5 *-.>. & 032 * E M 5<;�E J 8 2B=CE M ; =>EB0 5 W � =S;�;@8 = .C.�M 5<E:5 ,3* 8/5<6 %�& 8 2B5

E:5<4 M 5IE:5 ,/* 8 =>E M ; 5 80@1E * A 5 . &O%�& * 6:61=>E M @ \ W ; 8 J \ W ; a]W � "

% " L � JN. ? A E"$10 *N. =>E M H %'& * ?:EB=P8�=CE B L ( ��2:=S;�=S; A ? . 8 =>F . & =CE MD* 0 JN. ? A EU=>E � %'& * ?:E:=C8b\ 42:=S032 =>;

5���?:=p? *-. 5IE�8�8 J 8 ,3* EB; GKJN, A =CE M 8 2:5 M 5IE:5 ,/* 8 =>E M ; 5 8 J-G � GK,/J A . W 0V�$ $�$�VuW ; V $�$ $�VXW % 0 8 J . W� 0V�$ $�$�V \ W ; V�$�$ $�VXW�% 0

42:=S032 J-G 0 J ? , ; 5 % J 8/2 M 5IE:5 ,/* 8 5�8 2B5 ; * A 5�8/2:=CE M ;@=>EB0 5�\ Hg\ W ; L�c W ; "

0 " L � JL. ? A EQ� E�8/5 , 032 * E M 5 ( ��2:=S; =S; � ?B;@8 ; 4=P83032:=CE M 8�4 J 0 JL. ? A E ;=CER8/2:5 A * 8 , =RA � "�� ER8 5 , A ; J-G J ? ,

% * ;@=S; . W� V $�$�$4VXW ; V�$ $�$�VuW �KV�$ $�$�VuW % 0 @�4�5 *-, 5 � ?B;@8;@4=C8/032:=>E M W ; * EB6 W � @�42:=S032 J-G 0 J ? , ; 5 6 J 5<;�E J 8�032 * E M 5 � "

T 5+4= .>. 2 * EB6 . 5 *-.>. 8 2:57;@5�8/2 , 5<5 , J 4 J F 5 ,3* 8 = J E ; * 8 J EB0I5�=CED8 2B5+E:5�A'8 F *N,/*NMN,3* F:2 "

6 " L � J 4 � 5IF .S* 0I5 A 5IE�8

5�" L � J 4 $10 *N. =>E M H %'& * ?:EB=P8 =>E B L

G " L � J 4 � E�8/5 , 032 * E M 5

� J 4 J F 5 ,3* 8/= J EB; *-, 5 * % =C8 A JL, 5 0 J A F . =>0 * 8 576 8/2:5 & 032 * E M 5 5�?N5 , & 5 . 5 A 5<E�8 =CE"8 2:5 % * ; =>;�@ * EB6 0 * E

032 * E M 5 42 * 8 5 . 5 A 5<EL83; *N, 5 * 0 8 ? *-.>. & =CE � "R� J 4�5�?N5 , @ 42 * 8 4�5 4= .C. ;@2 J 4 =S;+8 2 * 8 B % � ����H B % L�=S; =S; J

A JN, F:2:=S0+8 J�B % � ����,H B % Lt@�42:5 , 5� =S; 8/2:5 5 . 5 A 5IE�8 *N, &DA * 8 , =RAQ0 JN,/, 57;@F J E 61=CE M 8 J 8/2:5 * F:F ,/J F , = * 8/5 , J 4

J F 5 ,3* 8/= J ED8 2 * 8 4 * ; F 5 , GKJN, A 5<6 J E � "�� J EB; =S615 , 8 2:5 A * F:F:=>E M % 5 8�4�5I5<EQ8 2:57;@5 8�4 J A J 61? . 5<; 615 H E:5<6 %'&

V

Page 17: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

� H F L(c F "�$'=>EB0 5 =S; * ER5 . 5 A 5IE�8 *-, & A * 8 , = AD=P8 =S;5 * ; & 8 J ; 5I5+=C8 2 * ; * EO=>Ei?N5 , ; 5 " � G �4 * ;�0 , 5 * 8 576 %'&

F 5 , GKJN, A =>E MR* ,/J 4 , 5IF .S* 0I5 A 5IE�8 J E 8/2:5 =>6:5IE�8 =C8 &QA * 8 , =RA @ 4�5 0 * E 0 , 5 * 8 5� %'& F 5 ,@GKJL, A =>E M 8 2:5 ; * A 5

,/J 4 , 5<F .S* 0 5 A 5IE�8 % ?18 %'& ?B; =>E M 8 2:5+E:5 ML* 8 =p?N5 J-G 8 2:5 JL, = M =>E *-. ;/0 *N.>*N, " � G 4 * ;�0 , 5 * 8 576 %'& ;/0 *N. =>E M * , J 4

%'& \ 8 2:5<E =S;�0 , 5 * 8/5<6 %'& ; 0 *-. =CE M 8 2:5 ; * A 5 ,/J 4 %'& \ " � G 4 * ; A * 6:5 %'& ;@4=C8/032:=>E M 8�4 J JNG 8 2B5

0 JL. ? A E ; JNG 8 2:5D=>6:5IE�8 =C8 & A * 8 , = Am@ 8/2:5IE %c " � J 4 0 J EB;@=S615 , � H��SLt@�;@=>EB0 5 =S; * A * 8 , = Am@ � =S; *

2 J A J A JN, F:2:=S; A @ * EB6 =>E G * 0 8�@';@=>EB0 5 2 * ; * E =>Ei?N5 , ; 5K@ � 2 * ; * E =CEi?L5 , ;@5 G ?:E 0 8 = J E � H F L(c F " ��2'?B;

� A ?B;@8 % 5 * E =CE � 5<0 8/= J E8@ % ?18�=P8�=>; *-. ; J * ; ? , � 5<0 8/= J E J E�8 J�B % ;@=>EB0I5�8/2:5 0 JN. ? A EB; JNG 8 2:5�=S615IE�8 =C8 & A * 8 , =RA��

; F * E B % @ * EB6 ; J 6 J 8 2:5�0 JL. ? A E ; JNG�* E 5 . 5 A 5IE�8 *N, & A * 8 , =RA �@1;@=>EB0 5 J EB5 0 JL. ? A E J F 5 ,3* 8 = J EO0 * EO032 * E M 5

� 8 J * E & " ��2B5 , 5 GKJN, 5 � H�SL�=S;�=>; J A JL, F:2B=>0 8 J � "���2'?B;�8 2:5�8/2 , 5<5 ,/J 4 J F 5 ,/* 8 = J EB; J E�� 8 * $N5 � 8 J * E

=S; J A JL, FB2:=>0 B A J 61? . 5K@ � � "

� J 4 4�5OE:5I576 8 J ;@2 J 4 8 2 * 8 B % � � =>; =S; J A JL, FB2:=>0 8 J_B % � � � * G 8/5 , F 5 ,@GKJL, A =>E M"*U, J 4 J F 5 ,/* 8 = J E

H , 5 A 5 A % 5 , 0 JN. ? A E J F 5 ,3* 8 = J E ; 032 * E M 5 E J 8/2:=>E M L "���2:5 A * F:F:=>E M 8 2 * 8 8 * $N57; F a � 8 J � H F L a � � c F a � �

4 JN, $';�" T 5 A ?B;@8 ;@2 J 4 8/2 * 8 8/2:5 A * F:F:=>E M =>; 4�5 .>. 615 H EB5<6 "�$'?:F:F J ; 5�8 2 * 8 F a � c � a � " ��2:5IE F @ � c >

GKJN, ; J A 5 > Y � @ E J 4 F @ � c > Y � H�_L�c � � " � 8 =S; * 2 J A J A JL, F:2B=>; A ; =CE 0 5 =C8 =S; * A * 8 , = A

8 ,3* E ; GKJL, A * 8/= J E8@�=P8=S; * E =>E � 5<0 8 = J ED;@=>EB0 5 �2 * ; * EO=CEi?N5 , ; 5K@ * E 6 =C8�=S; * ; ? , � 5<0 8 = J EO; =>EB0 5 � =S; * ; ? , � 5<0 8/= J E

J E�8 J�B % "+� J 4 4�5 $'E J 498 2 * 8�F 5 , GKJN, A =CE MO* H E:=P8/5 E'? A % 5 , JNG , J 4 * EB6 0 JN. ? A E J F 5 ,3* 8/= J EB; J E � 6 J 5<;

E J 8�032 * E M 5 B % � � ?BFO8 J =>; J A JL, F:2B=>; A "

T 5 4= .C. ;@2 J 4 8/2 * 8 M =p?N5<E * > � >DA * 8 , = A �I@ 4�5 0 * E , 5<6:?B0 5 �I@�?B; =CE M 8 2:5 * % J ?N5 J F 5 ,/* 8 = J EB;�@18 JO*

61= *-MLJ E *N. A * 8 , = Am@ * E 6 8/2'?B;�E J 8 *N. 8/5 , 8/2:5 0 JL, , 5<; F J EB61=>E M A J 61? . 5K@ * 8 . 5 * ;@8?:FD8 J =S; J A JL, FB2:=>; A "

��2:5 � .CMLJN, =P8/2 A GKJL, �5<61? 0 =>E M J ? , * 8 , = A (

V " � * $L5 � ;�� � 8 J % 5 * E & E J E! �<5 ,/J 5 . 5 A 5<EL80@8HK= G 8/2:5 , 5 =S;�E J ; ?B032O5 . 5 A 5IE�8�8 2:5 A * 8 , =RA =>;�61= *-MLJ E *N. L J-G

8 2B56� �I� A * 8 , =RA ��42:5 , 5bosH � ;�� � L reo�H�4L GKJN,�* E & E J E �<5 ,/J 5<EL8 , & ��=CE J ? , A * 8 , =RA � "���2 * 8�=S; � ;�� � =>;

; A *N.C. 5<;@85IE�8 , & JNG 8/2:5 A * 8 , =RA ?:FD8 J 8 2B5 G ?:EB0 8/= J E�o "

� G�� ;�� � 61=p?'=>6:5<; 54?L5 , & 5 . 5 A 5IE�8 =CE =C8/; ,/J 4 * EB6 0 JL. ? A E8@8 2:5<E ? ;@5Q0 JL. ? A E , 5IF .S* 0I5 A 5IE�8 * EB6 , J 4

, 5IF .S* 0I5 A 5IE�8 8 J A * $L5+8 2 * 8 ,/J 4 * EB6 0 JN. ? A E �I5 , J HK5�A:0I5IF18 GKJN,�� ;�� � L�" T 5 2 * ?N5 5 ��5<0 8/=R?L5 . & , 5761?B0 576R8 2B5

61= A 5IEB; = J E J-G 8 2B5 A * 8 , =RA %'& V , J 4 * EB6 V 0 JN. ? A E " � J 8 J ;�8/5IF V 4=C8 2 J ? , E:5<4O� @ V �I� @ V A * 8 , =RAm@

V �

Page 18: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

?:E . 5<;/; J ? , A * 8 , = AO2 * ; J E . & J E:5+5 . 5 A 5<E�8�@1=>EO8/2 * 8�0 * ; 5�F , J 0I5I5<6 8 J ;@8 5<F � "

��8/2:5 , 4=>; 5 � ;�� � 6 J 57;@E�� 8�6:=R?'=S615�; J A 5 5 . 5 A 5<EL8 J-G =C8/; , J 4 JN, 0 JN. ? A E8@'; * & =P8�6 J 5<; E�� 8�61=p?'=>615 ��� @�8 2:5<E%'& 8 2:5+61=p?'=>; = J E *-.>MNJN, =C8 2 A GKJN,* � ?B0 . =S615 * EO� J A * =CE8@'8/2:5 , 5�5�A1=>;@8/;�E J E! �I5 , J 5 . 5 A 5IE�8/; \ * EB62,R; ?B032 8 2 * 8

� � c , � ;�� � a \i@:42:5 , 5bo�Hg\<L � osH � ;�� � L " ��2:5IE * 6:6 @ ,D8 = A 5<;�8 2B5 , J 4�H 0 JL. ? A EsL�0 J E�8 * =CEB=CE M�� ;�� � 8 J 8 2B5,/J 4 H 0 JN. ? A EsL+0 J E�8 * =CE:=>E M�� � "O��2:5<E 8 2:5 5<EL8 , & * 8 � � % 5<0 J A 5<; \ "R��2:5 ; A *-.>. 57;�8 5 . 5 A 5IE�8�HK?:F 8 J 8 2B5G ?:EB0 8 = J E�oiL�=>;E J 4 * 8 . 5 * ;�8E J % = MNM 5 , 8 2 * E \i@ * EB6D4�5+0 * E , 5<;@8 *-, 8 ;�8/5IF V "

��2:=S; HB, ;�8�F , J 0I5<;/; 0 * E:E J 8�0 J E�8 =>E'?:5 =>EB615 H E:=C8 5 . & ; =CE 0 5 8 2:5 A * 8 , =RAR2 * ; H E:=C8 5 61= A 5<EB; = J E * E 6 ;@=>EB0I5

* 8 5 * 032 ;@8 5<F =>ER8 2:5 ;@570 J E 6Q0 * ; 5�o�Hg\<L M 5 83;�; A *-.>. 5 , @ ; J 5�?N5IE�8/? *-.>. & o�Hg\<LOc osH V L�HK=>E 42:=S032U0 * ;@5b\ 4 J ? . 6

% 5 * ?BE:=P8 * EB6D=P8 4 J ? . 6D61=p?�=S615+5�?N5 , & 8 2:=>E M =CER=C8/; ,/J 4 * EB6R0 JN. ? A EsL "

� E A * E & 4 * & ;+8/2:=S; F , J 0 5761? , 5 =S; ;@= A = .S*-, 8 J 8 2:5R� ? 0 . =>615 * E � .>MNJN, =C8 2 A @�5�A:0I5IF18 8 2 * 8 4�5 *-, 5 E J 8

E:570 5<;/; *-, = . & F 5 ,@GKJL, A =>E M J F 5 ,3* 8/= J EB; J EO8�4 J 5 . 5 A 5<EL83;�"

� " T 5 E J 4 2 * ?N5+; J A 5�F 5 , A ?:8 * 8/= J E J-G * 61= *NMNJ E *-. A * 8 , =RA "#�; 5 0 JN. ? A ED=>E�8 5 , 032 * E M 5 8 J 0 J E ;�8 , ?B0 8

* 61= *-MNJ E *-. A * 8 , =RA "

��2:=S; F ,/J ?N57; 8 2 * 8 42:5IE B =S; * � ? 0 . =>615 * E 6 J A * =CE8@�54?L5 , & H E:=C8 5 . & M 5<E:5 ,3* 8/5<6 B A J 61? . 5R0 * E % 5

5�A1F , 5<;/; 5<6 * ;8/2:5+61= , 570 8�F , J 61? 0 8 J-G 0 & 0 . =>0 B A J 61? . 5<;�@'42:=>032 *-. ; J F ,/J ?N5<;8 2 * 8�54?N5 , & H EB=P8/5 . & M 5<E:5 ,3* 8/5<6

MN,/J ?BFO0 * E % 5+54A'F , 57; ; 5<6 * ;�8 2B5 61= , 5<0 8F , J 61? 0 8 J-G 0 & 0 . =S0 MN,/J ?:FB;�"

��� +������� �fPQ1 T 5 4= .>. E J 4 ; 2 J 4 8 2:=S;�8 2:5 JN, 5 A =CE ? ;@5 J E * H E:=C8 5 . & M 5IE:5 ,/* 8 5<6R� % 5 . = * E ML, J ?:F

H * � A J 61? . 50L " � 5 8 8 2:5 M 5IEB5 ,3* 8 =>E M ; 5 8 GKJL, � % 5

�� ���� �V

� �� @ VN�

� �� �@ �@

�� � ; J JL, A * 8 , =RA =S; Z c

�� � @ V � N @ �V � @

�$ � G 4�5 ?B;@5IosH F L�c � F � @ 8 2:5 ; A *-.>. 57;�8 5 . 5 A 5IE�8 JNG 8 2:5 A * 8 , = Am@ ?:FQ8 J oq@ =S;�H @ V L J E , J 4

J EB5 0 JL. ? A E 8�4 J @�HK8 2:5 V J E , J 4 8/2 , 5<5 0 JN. ? A E J EB5 4 J ? . 6 4 JL, $ *-. ; J L�" @ V 6:=R?'=S615<;�54?L5 , & 8/2:=>E M =>E =C8/; , J 4

* E 6 0 JL. ? A E ; =CE 0 5 =C8 =S; * ?:E:=C8�@ 8 2'?B;�4�5 2 * ?N5 Z c �� � @ V � N @ �W N V

�* G 8/5 , F 5 ,@GKJL, A =>E M 8 2B5 * FBF , J F , = * 8 5

,/J 4 , 5<F .>* 0 5 A 5<E�8 J E , J 4 8/2 , 5I5 " * $'=>E M 8/2:5 , 57;�8 J-GB,/J 4 J EB5 �<5 ,/J =S; * 032:=>54?N576 %'& F 5 ,@GKJL, A =>E M 8�4 J 0 JN. ? A E

, 5IF .S* 0I5 A 5IE�8 J F 5 ,/* 8 = J EB; ; J Z c �� N @ V N N @ �W N V

�$ � J 4 5 ��5<0 8 =p?N5 . & 4�5 0 * E GKJL, M 5 8 * % J ?18 ,/J 4 J E:5 * EB6

0 JL. ? A EO8�4 J J-G 8 2:5 A * 8 , =RAm@ . 5 * ?�=>E M ?B;�4=P8/2 Z � c�� @ �W V �� $ � ?:8 GKJL, 8 2:5+; * $N5 J-G ?�=S; ? *-. ; = A F . =>0I=P8 & . 5I8

?B; . 5 * ?L5�8 2 * 8 , J 4 * EB6 0 JL. ? A E =>E8@ *-. 8 2 J ? M 2 4�54 J E � 80 J EB; =>6:5 , 8 2B5 A GKJN, * E & J F 5 ,3* 8 = J E ;�" ��2�? ; 8 2:5 EB5I4

V J

Page 19: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

; A *N.C. 5<;@8�5 . 5 A 5IE�8 J-G Z =S;�H @ �KL J E , J 4 8�4 J 0 JN. ? A EU8/2 , 5I5 " T 5 0 * E 8 2B5IE ?B;@5 0 JN. ? A E , 5IF .S* 0I5 A 5IE�8 J E

0 JL. ? A E J E:5 8 J M 5 8�8 2:5 E:5I4 Z c �� N @ V NV N @ ���� N V

�@ * EB6R8/2:5IE"F 5 ,@GKJL, A 0 JN. ? A E , 5<F .S* 0 5 A 5IE�8 % * 0 $ J E

0 JL. ? A ED8/2 , 5<5�8 J M 5 8 Z c �� N @ V NV N N��� N J �

�@ * EB6O8/2:5IEQF 5 , GKJN, A =CE M ,/J 4 , 5IF .S* 0I5 A 5IE�8 *NML* =>EO8 J M 5I8 Z c

�� N @ V NV N NN N J �

�$ T 5 E J 4 J E . & 2 * ?N5 8 J ?B;@5 0 JL. ? A E"=CE�8/5 , 032 * E M 5 * EB6 4�5 2 * ?N5 Z c �� V N N

N @ V NN N J �

�"

��2'?B; J ? ,�H E:=C8 5 . & M 5IE:5 ,/* 8 576 ML, J ?:F =S;=S; J A JL, FB2:=>0�8 J � � ' V )!P � � ' @ V )!P � � '>J �9) 42B=>032R=S; � ?B;�8������ "

V �

Page 20: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

*,+.-0/(C 2 �� � � +.;K/�5s-6= � +./�=Q5 3 554 � = 3 =g/ � � � � � 3 � -6+`/ � 7 $ 587 9 ����I=�� '!3 =��m9 �

� J 4 8 2 * 8+4�5 2 * ?N5 ; 2 J 4EQ8 2 * 8+4�5 0 * E G * 0 8 JN,�* E & B A J 61? . 5 =>E�8 JQ* 61= , 5<0 8�F , J 61?B0 8 J-G 0 & 0 . =S0 B

A J 61? . 5<;�@N4�54= .>. ; 2 J 4 8/2 * 88 2:=S; G * 0 8 JL, = � * 8 = J E =>; ?:E:= �L?B5 42:5IE 4 , =P8 8 5<E =CE * 0I5 , 8 * =>E GKJN, A " T 54= .>. ; 2 J 4

8 2 * 8�54?N5 , & H E:=C8 5 . & M 5IE:5 ,/* 8 5<6 B A J 6:? . 5M@�42:5 , 5 B =S; * EU� ?B0 . =S615 * E 6 J A * =CEf@�0 * E % 5 4 , =C8@8/5IE ?:EB= ��?:5 . &

HKE J 8 0 J ?:E�8 =>E M 8/2:5 JN, 615 , =>EQ42:=S032R8 2:5 & *N, 5+4 , =P8 8 5<EsL * ; * 6:= , 5<0 8�F ,/J 6:?B0 8 JNG B � ;4@ * EB6 J-G B A J 61? . 5<; J-G

8 2B5 GKJL, A B � ' T % ) 42:5 , 5 T =>;�= , , 5<61? 0 = % . 5�"���= , ;�8�4�5�4= .>. ;@2 J 4 8/2 * 8�8 2B5 E'? A % 5 ,�JNG G * 0 8 JL, ; J-G 8/2:5 GKJL, A B

=S;�?:E:= �L?B5 "

���b�����"$ CfP>1 � E 8/2:5 * % J ?N5 G * 0 8 JL, = � * 8 = J E8@ 8/2:5E'? A % 5 ,�JNG G * 0 8 JN, ; J-G 8 2:5 GKJL, A B � .0N 0 c B =>;

?:E:= ��?:5 " *�-658554 2 $1?:F:F J ; 5 B =S; * � ?B0 . =S615 * E 6 J A * =>E * E 6+8 2 * 8D� =S; * H EB=P8/5 . & M 5<E:5 ,3* 8/5<6 B A J 61? . 5M@ 8 2:5<E

%'& 8/2:5 JL, 5 A " q@M� =S; =S; J A JL, FB2:=>0�8 J�B % PCB � ' T! )&P � ��� P�B � ' T � ) @'42:5 , 5 T ; =S; E J E! �<5 ,/J @LE J E! ?:E:=C8�" � . ; J; ?:F:F J ; 5�8 2 * 8�� =S;�=>; J A JL, F:2B=>0�8 J B � P�B � ' W ) P ��� � P�B � ' W�� ) 42B5 , 5 W ;�=>; E J E! �I5 , J @ E J E ?:EB=P8 " ��2:5 , 5 GKJN, 5

B % PB � ' T )*P ����� P�B � ' T � ) =S; =>; J A JN, F:2:=S08 JbB � P�B � ' W �) P ��� � P�B � ' W�� ) " � J 4 ;@=>EB0 5 B =S; *-. ; J�* E =>EL8/5 ML,/*N.

6 J A * =CEf@ B % P]B � ' T ) P � ��� P]B � ' T � ) 2 * ; * . =CEB5 *-,/. & =>EB615IF 5IE 615IE�8 ; ? % ; 5 8 JNG ; = �I5 > @ E * A 5 . & 8 2:5 H , ;@8 >0 JL. ? A E ;�H JL,�,/J 4 ;XL J-G 8 2:5 > a�� =>6:5IE�8 =C8 &RA * 8 , = Am@ V�$ $�$�V % "���2:=S;�; 5 8�=>; . =>E:5 *-,/. & =CEB6:5IF 5IEB615<E�8�; =CE 0 5 B

2 * ; E J �<5 ,/J 61=p?�=S; JN, ;�" T 2 * 8 4�5 4= .>. E J 4 ; 2 J 4 =S;8/2 * 8 * E & .S*-,/M 5 , ;@? % ;@5I8 =S; E J 8 . =CE:5 *-,/. & =CE 615IF 5IEB6:5IE�8�"

$'?:FBF J ;@5Q8/2 * 8 �� V�$ $�$�V � %�� =>; * . =>E:5 *N, . & =>EB615<F 5<EB615<EL8R;@? % ;@5I8 J-G B % P B � ' T! ) P ����� P B � ' T � ) @�8 2:5<E. 5 8 T c T! �TM" �����XT � @ * EB6 0 J EB;@=S615 , 8/2:5 ; 5 8 T �� 6V $�$ $�V T � %�� " � . 5 *-,/. & 8/2:=S;D; 5 8 A ? ;�8 *-. ; J % 5 . =CE:5 *-,/. &

=>EB615IF 5IE 615IE�8�@ % ?18 =C8 *-. ; J A ?B;@8 % 5 * ; ? % ; 5 8 JNG 8/2:5 A J 61? . 5 M 5IE:5 ,/* 8 5<6 %'& V�$ $�$4V % @ ; =>EB0 5 8/2:5 .S* ;@8 �

; ? A A * EB6:; A ?B;@8 *-.>. % 5 �<5 ,/J " ��2:5 , 5 GKJN, 5K@ %'& 8/2:5 JL, 5 A �!" .@ � V�$ $�$�V � %�� =>; . =CEB5 *-,/. & 615IF 5IE 615IE�8�@ 42:=S032

=S; * 0 J E�8 ,/* 61=>0 8 = J E8@���2�? ;4@ 8/2:5 .S*-,/M 5<;@8 . =>E:5 *-,/. & 615<F 5<EB615<EL8 ;@? % ;@5I8 JNG B % PeB � ' T )/P ��� � P B � ' T � ) 2 * ;; =��<5 > " $1= A = .>*N, . & @ B � P B � ' W� ) P ��� � P B � ' W � ) 2 * ; * . =>E:5 *-,/. & =CE 615IF 5IEB6:5IE�8�; ? % ; 5 8 J-G ; =��<5�� @ * E 6R54?N5 , &

.S*-,/M 5 , ;@? % ;@5I8 =S; . =CEB5 *-,/. & 615<F 5<EB615<EL8 " $'=>EB0I5 8 2B5 8�4 J A J 61? . 5<; *-, 5 =>; J A JL, F:2B=>0K@�=C8 =S; 5 * ; & 8 J ; 5I5 8 2 * 8

B % P B � ' T! ) P ����� P B � ' T � ) 2 * ; *. =CEB5 *-,/. & =>EB615IF 5IE 615IE�8�;@? % ;@5I8 J-G ; =��<5 � * E 6�8 2 * 8 B � P B � ' W� )KP ��� � PbB � ' W � )2 * ; * . =>E:5 *N, . & =>EB615IF 5IE 615IE�8�; ? % ; 5 8 J-G ; =��<5 > " ��2�? ; > c � "

T 5 4= .C.�*-. ; J F ,/J ?N5 8/2 * 8 42B5IE B % P�B � ' T! )KP ��� � P�B � ' T � ) =>; =>; J A JN, F:2:=S0 8 J"B % P�B � ' W )?P ����� P�B � ' W � ) @V �

Page 21: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

8 2 * 8 B � ' T& ) P � ��� P B � ' T � ) =S;=S; J A JL, FB2:=>0�8 J�B � ' W� ) P � ��� P]B � ' W � ) ���b�����"$ C P � � GfB =S; * E �?B0 . =>6:5 * EO6 J A * =>E * E 6 8 2B5 B A J 61? . 5K@&� c B % PSB � ' T! ) P � ��� P

B � ' T � ) =>;�=>; J A JL, F:2B=>0�8 J # c B % PSB � ' W� ) P ��� � PSB � ' W � ) 8 2B5IE B � ' T! ) P ��� � PSB � ' T � ) =S;�=>; J A JN, F:2:=S0�8 JB � ' W )/P ����� P B � ' W�� ) " *�- 5f5 4X2 $'=>EB0 5 8 2B5 8�4 JO*-, 5 =S; J A JN, F:2:=S0�8 2:5 , 5 5�A1=S;�83; * E =>; J A JL, F:2B=>; A % 5 8�4�5I5<E

8 2B5 A @ . 5I8 ?B; 0 *-.>. =P8 � " � J 4 0 J EB; =S615 ,�� c . H F- V F " V�$�$ $4V F5%�� � Y � � F- c ����� c F % cjN 0 @�E J 8/=>0I5

8 2 * 8 � =>; * ; ? %:A J 61? . 5 JNG � " ��EB6 0 J EB;@=S615 , 42 * 8 2 * F:F 5IEB; 42B5IE & J ? , 5<;@8 , =S0 8 � 8 J�� " T 5 4= .C.

0 *N.C. 8/2:=S; A * F:F:=>E M 8 2 * 8�8 * $L5<; � 8 J �"H � L�@ � " � J 8 =S0 5 8 2 * 8 � =S;+;�8/= .>.* E"=CE � 5<0 8/= J E " T 5 4= .C. ;@2 J 4!8 2 * 8

� H � LOc � c .iH � 6V � "KV�$�$ $�V � %�� � Y #�� � c ��� � c � % c N 0 "� 8 =S;�5 * ; & 8 J ;@5<5 8/2 * 8 � H � L =S;�0 J E�8 * =>E:5<6R=>E

�U@ GKJN, ; ?:F:F J ; 5 8 2 * 8 GKJN, ; J A 5 F Y � @ 4�5 2 * ?N5 8/2 * 8 � H F L,c H�� 6V $�$�$4V � �KV�$ $�$�V � % V � %�� <V�$�$ $�V � %�� � L�@ 42:5 , 5

� � =S;�E J 8+5���? *N. 8 J �<5 ,/J " ��2:5<E . 5 8 T c T! �TM" �����XT � @ ; J N�c � H>NML"c � H T F L�c T � H F L�c H T � V $�$�$4V T � �KV�$ $�$ L42:=S032U=S;�E J 8�5���? *N. 8 J �I5 , J ;@=>EB0 5 T � � =S;�E J 8+5���? *N. 8 J �I5 ,/J % 570 * ? ;@5 B =>; * EU=>E�8 5 ML,/*N. 6 J A * =>E " � ?18�4�5

0 * E *N. ; J ; 2 J 4 8/2 * 8 � A * FB; ��J E�8 J � " � = , ;�80@ . 5I8�? ;�6:5 H E:5 W c W W " � ��� W�� "�� J 4 = G 4�5�8 * $L5 * E & � Y �U@

8 2B5IE 8/2:5 , 5�54A'=S;@8/; F Y � ; ?B032 8/2 * 8 �"H F L�c � "�� J 4 ; ?:F:F J ; 5 8/2 * 8 F �Y � @L8/2:5IEO; J A 5 5IE�8 , & J-G8F @1; * & F ; @

=S; E J 8 5���? *N. 8 J N.@ 42:5 , 5 V r = r > " ��2'?B; W F =>; E J 8�5���? *N. 8 J N "�$ J ��H W F L�c W �"H F L�c W � c Nlc �"H>NMLt@

42:=S032 =S; * 0 J E�8 ,3* 61=S0 8/= J E ;@=>EB0I5 ��=S; J E:5 8 J J E:5 " ��2'?B; 8 2:5 A * F:F:=>E M$� ( � � � =S; * E =>; J A JL, F:2B=>; A @

42:=S032Q0 J A F . 5 8/5<;�8/2:5+F ,/J�JNG "

� J 494�5 4= .C. F ,/J ?N5�8 2 * 8 * E & B A J 61? . 5 JNG 8 2:5 GKJN, A B � ' T ) =>; =S; J A JL, FB2:=>0�8 J * E B A J 6:? . 5 J-G 8 2B5

GKJN, A B � ' T! ���� ) P � ��� P B � ' T&%���� ) @ 42B5 , 5 T ; =S; * E = , , 5<61? 0 = % . 5 5 . 5 A 5<E�8 J-G B " � = , ;@8+4�5 4= .C. F , J ?L5 8�4 J

; A *N.C. . 5 A A * ;�"

� � � � +h1�2 � 5 8 B % 5 * F , =CEB0I=CF *-. =S615 *N. 6 J A * =CE * EB6 . 5 8 F V � Y B " $1?:F:F J ; 5 F * EB6 � 2 * ?L5 E J

0 J A A J E�E J E! 8 , =R?'= *-.�G * 0 8 JN, ; 4�5D4= .>. F ,/J ?N5 8/2 * 8 8/2:5 , 5D5�A1=>;@8 ^ * E 6 � =>E B ;@? 032�8/2 * 8 ^ F a � � c V "

� J E ;@=S615 , ' F5)�* E 6 ' �&) ; =CE 0 5 B =>; * F , =CE 0 =>F *-. =S615 *N. 6 J A * =>E8@ 4�5 $�E J 4!8 2 * 8(' T ) c ' F V �&) c ' F ) a ' �!) "

��2:5<E8@ ;@=>EB0I5 F V � Y ' F V �&) @ 4�5 2 * ?L5 8/2 * 8 F c � T * EB6 � c:� T GKJN, ; J A 5 � V � Y B @ ; J T =S; * G * 0 8 JN,�J-G

% J 8 2 F * EB6 � @'8/2�? ; T =S; * ?BE:=P8 " ��2:5 , 5 GKJL, 5N' F V �!) c B @ * EB6D;@=>EB0I5 V Y B @'8 2:5 , 5�54A'=S;@8 ^MV �(Y B ; ?B032O8 2 * 8

^ F a � � c V "

��2:=S;E:54A'8 . 5 A A * =S; , 5 A =>E:=S; 0I5IE�8 J-G 8 2:5 ��2B=CE:57;@5 � 5 A * =>EB615 , ��2:5 JL, 5 A "

V �

Page 22: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

� � � � +��m2 � 5I8 B % 5 * F , =>EB0I=CF *N. =>615 *-. 6 J A * =CE * E 6 . 5 8 T! V T#" % 5 E J E! �I5 , J E J E! ?:E:=C8�5 . 5 A 5IE�8/;

J-G B * EB6 *N. ; J ; ?:F:F J ; 5 8/2 * 8 T +* EB6 T " 2 * ?N5 E J 0 J A A J E E J E! 8 , =R?'= *-.�G * 0 8 JN, ;�" T 5 4= .C. F ,/J ?N5 8 2 * 8 GKJL,

* E &_TsVuWY B @ 8/2:5 , 5 5�A1=S;�83; F Y B ;@? 032"8/2 * 8 F a ' T ) c T�a ' T )+* EB6 F a ' T "<) c W a ' T "<) " � & 8 2B5

F , 54?'= J ?B; . 5 A A * 8/2:5 , 5 54A1=>;@8/; ^�� Vu^��4V ��� V ���"Y B ; ?B032R8 2 * 8 ^�� T a ��� T " c T * E 6 ^�� T a ���XT " c W "���2:5<E

. 5 8 F c ^ � T! da � � TM" "�� . 5 *-,/. & 8 2:=S;=S; * ; JN. ?18/= J E 8 J 8 2:5 ; & ;�8/5 A "

���b�����"$ C P�� � G:B =S; * �?B0 . =S615 * EO6 J A * =CEf@L8/2:5IE * E & B A J 61? . 5 J-G 8/2:5 GKJN, A B � ' T ) @142:5 , 5

T =>; * E J E! �I5 , J E J E! ?:E:=C8 5 . 5 A 5IE�8 JNG B @ =S;+=>; J A JL, F:2B=>0 8 JCB � ' T! ) PhB � ' T#" ) 42:5 , 5 T c T! tT#" * EB6 T!

* E 6 T#" 2 * ?L5 E J 0 J A A J E G * 0 8 JN, ;�" *�-658554 2 � 5 8 B % 5 * �?B0 . =S615 * E�6 J A * =CE * EB6 . 5 8 ?B; 0 J E ;@=S615 , 8 2B5

2 J A J A JN, F:2:=S; A � ( B � B � ' T! ) P B � ' T#" ) @42:5 , 5 T& * EB6 T#" *-, 5RE J E! �I5 , J E J E! ?:E:=C8 5 . 5 A 5IE�8/; 8 2 * 8

2 * ?L5OE J 0 J A A J E G * 0 8 JN, ; * EB6 � H F L!c�H F a ' T! ) V F a ' T#" ) L�" � . 5 *-,/. & 8 2:=S; A * F:F:=>E M ; * 8 =S; H 5<; *-.>. 8 2B5

2 J A J A JN, F:2:=S; A F , J F 5 , 8/=C57;�" T 5 E:5<5<6O8 J F ,/J ?N5 8 2 * 8 � =>; ; ? , � 5<0 8 =p?N5 "� G 8 * $L5 * E & H Tba ' T �) VXW a ' T " ) L Y

B � ' T �) P B � ' T " ) @ %�& 8 2:5 ;@570 J EB6 . 5 A A * 8/2:5 , 5 5�A1=S;�83; F Y B ;@? 032 8 2 * 8 F a ' T ) c T�a ' T ) * EB6

F a ' T "<) c W�a ' T "<) @ 8 2'?B; � H F LIc H TIa ' T ) VuW�a ' T "<) L�" T 5 *N, 5 MNJ =CE M 8 J ?B; 5 8 2B5 HB, ;�8 =S; J A JN, F:2:=S; A

8 2B5 JN, 5 A " � 5I8 T! 4T#" c T " T 5�E:5<5<6 8 J F ,/J ?N58 2 * 8 � 0\ � c:' T ) " $'?BF:F J ;@5�8/2 * 8 � H F L(c�HQN a ' T! ) V N a ' T#" ) L�"

��2:5<E F =S;+=>E�' T! )�* EB6U8 2'?B; * A ? . 8/=CF . 5 JNG T! "D$'= A = .>*N, . & F =S; * A ? . 8 =>F . 5 J-G T#" " $'=>EB0I5 B =S; * ?:E:= �L?B5

G * 0 8 JL, = � * 8 = J EU6 J A * =>E * E 6U; =CEB0I5 T! * E 6 T#" 2 * ?N5 E J 0 J A A J E G * 0 8 JN, ;4@ T � F @ � 0\ � =S;�0 J E�8 * =>E:5<6U=CE ' T ) "

� J 4 8 * $L5 F Y ' T ) @�8/2�? ; F c � T c � T! �TM" " ��2:5IE � H F L c H � T#"0T& fa ' T! ) V � T& �T#":a ' T#" ) L�ceN "���2�? ; %'& 8 2B5

HB, ;�8=S; J A JN, F:2:=S; A 8/2:5 JL, 5 A GKJN, A J 61? . 5<;�@ B � ' T ) P]B � ' T " ) =S;�=S; J A JL, FB2:=>0�8 JB � ' T ) "

T 5 0 * E E J 4!032 * E M 5 8/2:5 G * 0 8 JN, =�� * 8 = J ER8/2 * 8�4�5 GKJ ?BEB6 =>E 8/2:5 JL, 5 A " =CE�8 JR* G * 0 8 JN, =�� * 8 = J E JNG 8 2B5

GKJN, A B � , � � P � ��� P B � , % � � 42:5 , 5N, ; =>; = ,/, 5761?B0 = % . 5K@ * EB6 � ; � N " T 5Q4= .>. E J 4 ;@2 J 4 8 2 * 8 GKJL,

* E & H E:=C8 5 . & M 5IE:5 ,/* 8 576 A J 61? . 5 8 2B5 , 5 =>; J E . & J E:5 G * 0 8 JN, = � * 8/= J E JNG 8/2:=S; GKJN, A ?BF 8 J =S; J A JL, FB2:=>; A * EB6

, 5 JN, 615 , =>E M 8/2:5 G * 0 8 JL, ;�"

���b�����"$ C P C � G B =S; * 0 J A A ?18 * 8 =p?N5R?:E:=C8 *N, & , =>E M * E 6 � =S; * E B A J 61? . 5 * E 6 � =>;

=S; J A JL, FB2:=>0 8 J � DP � ��� P � % 42:5 , 5 � 8 2 ,/J ? M 2 � % *-, 5 *-. ; J�B A J 61? . 57;4@'8/2:5IER8/2:5 ;@? %:A J 6:? . 5 JNG � @

\9� c1.�\ F � F Y ��0 HG, =S; * E 5 . 5 A 5<EL8 J-G �,L =>;�=S; J A JL, FB2:=>0�8 J \ � 9P ��� � P \ � % " *�- 5f5 4X2 � = , ;@8 4�5 A ?B;@8 ; 2 J 4

8 2 * 8 \*� =S; , 5 *-.>. & * ; ? %:A J 61? . 5 JNG � 0 . 5 *N, . & =C8 =>;�E J E 5 A F:8 & ; =>EB0 5(\ N�c N " ��2:5IE 8 * $L5 * E & \ T�V \ W�Y \*� "

V<W

Page 23: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

��2'?B;"\ T�@ \ W c \`H TN@]W L Y \9� @ * EB6 GKJN, * E & ^lY B @ ^ \ T c \`H ^�T L Y � "���2'?B;,\9� =S; * ;@? %BA J 61? . 5 J-G

� " � J 498 J * 0 8 ? *N.C. & F , J ?L5�8 2:5 . 5 A A * " � * $L5 * E & \ T Y \9� HK42:5 , 5 T =S; � ?B;@8 * E *-, % =C8 ,3*-, & 5 . 5 A 5<EL8 J-G

� ; J T c H�� V�$�$ $�V � % L Lt@'8 2B5IE!\ T ce\`H�� V�$ $�$4V � % L(c Hg\�� V $�$ $4V \�� % L Y , � P � ��� P , � % "$'= A = .S*-,/. & @�8 * $L5

* E & Hg\�� V $�$�$�V \�� % L Y \ � P ��� � P \ � % @�8 2:5<E Hg\�� V $�$ $4V \�� % L�=S;�=>; J A JN, F:2:=S0�8 J \`H�� V $�$�$�V � % L�c \ T�Y \*� "

��2:5 , 5 GKJN, 5"\9� c \ � P ��� � P \ � % "

T 5 4= .C. E J 4 ; 2 J 4 2 J 4 4�5 0 * E , 5761?B0I5 8 2:5 F , J % . 5 A 8 J ;@2 J 4=CE M 8/2 * 8 8/2:5 , 5 =S; J E . & J E:5 4 * & 8 J

4 , =P8/5 B � ' , � � ) P � ��� PSB � ' , � �*) HK42:5 , 5�,O=S;= ,/, 5761?B0 = % . 50L�@ %'& ; 2 J 4=CE M 8 2 * 8�= G � 2 * ;�E J 0 J A A J E G * 0 8 JN, ;

4=C8 2N,R8 2:5<E�� B � 'G, � ) =S;�=>; J A JN, F:2:=S0 8 J�B � ' , � ) "

���b�����"$ C P�� � GfB =S; * � ?B0 . =S615 * E 6 J A * =CE * EB6B, V � Y B 2 * ?L5E J 0 J A A J E G * 0 8 JL, ; 8 2:5<E 8 2B5

B A J 61? . 5�� B � 'G, ) =>;�=S; J A JL, FB2:=>0 8 J�B � 'G, ) " *�-658554X2 � 5I8 B % 5 * �?B0 . =S615 * E 6 J A * =CE * EB6 ; ?:F:F J ; 5 , V � Y B

2 * ?L5 E J 0 J A A J E G * 0 8 JL, ;�" � J 4 0 J E ;@=S615 , 8/2:5 A * F:FB=CE M � ( B � � B � 'G, ) 615 H EB5<6 %'& � H F L c � F a ' , ) "

� . 5 *-,/. & 8/2:=>; A * F:FB=CE M =>;O4�5 .C. 615 H E:5<6 "���2:5 G * 0 8O8 2 * 8 � =S; J E�8 J�* EB6 ; * 8 =S; H 57; 8/2:5U2 J A J A JN, F:2:=S; A

F ,/J F 5 , 8 =>5<;�=S;�8 , =R?'= *-. * E 6U4= .C. % 5 J A =C8@8/5<6 "O� J EB;@=S615 , 8/2:5 $N5 , EB5 . J-G 8 2:=S;+2 J A J A JN, F:2:=S; A " � J 8/=>0I5 8 2 * 8

N a 'G, ) c � H F L cL� F a 'G, ) = G * EB6 J E . & = G , � � F "���2:5IEQ; =CEB0I5 B =S; * ?:EB= ��?:5 G * 0 8 JN, =�� * 8 = J ER6 J A * =CE * EB6�,

* E 6 � 2 * ?L5 E J 0 J A A J E G * 0 8 JL, ;�@#, � F " ��2:5 , 5 GKJN, 5 8/2:5 $N5 , E:5 .�JNG � =>;N' , ) " $ J %'& 8/2:5 H , ;@8�=S; J A JN, F:2:=S; A

8 2B5 JN, 5 A B � 'G, ) =S;=S; J A JN, F:2:=S0 8 J � B � 'G, ) "

T 5 4= .>. E J 4 8 2 * 8�8/2:5�8/2:5�;@= �I5 J-G 8/2:5 A =>E:= A ? A M 5IE:5 ,/* 8 =>E M ; 5 8 J-G 8/2:5 B A J 6:? . 5M@ B � 'G, � � ) P ��� � P

B � ' , � � ) =>; > H 42:5 , 5 ,D=S; * EO= , , 5<6:?B0 = % . 5�5 . 5 A 5IE�8 J-G B L "

���b�����"$ CfP�! � G8B =S; * E � ?B0 . =S615 * E 6 J A * =CE * EB6Q, =S; * E = , , 5<6:?B0 = % . 5 5 . 5 A 5IE�8 JNG8B @N8 2:5<E 8 2B5

; =��<5 J-G 8 2:5 A =CE:= A ? A M 5IEB5 ,3* 8 =>E M ; 5 8 J-G 8/2:5 B A J 61? . 5K@ � c B � 'G, � � ) P � ��� P B � 'G, � �*) =S; > " *�-658554X2

$'?:FBF J ;@5 8 2 * 8 B =S; * �?B0 . =S615 * EO6 J A * =CE * EB6 8 2 * 8/,D=S; * E = , , 5<61? 0 = % . 5�5 . 5 A 5<EL8 J-G:B "�� . ; J ; ?:F:F J ; 5 8 2 * 8

8 2B5 A =CEB= A ? A M 5IEB5 ,3* 8 =>E M ; 5 8 GKJN, 8/2:5 B A J 6:? . 5M@ � c B � 'G, � � ) P ��� � P B � ' , � � ) @ HK42B5 , 5 � ; � N Lt@ =>;

; A *N.C. 5 , 8/2 * E > " T 5 0 * E 54?L5IE ;@?:FBF J ;@5�=C8 =S; ; =��<5 >D@ V ;@=>EB0 5�4�50 * E *-. 4 * & ; * 6:6 5 . 5 A 5IE�8/; 8 J * M 5IE:5 ,/* 8 =>E M

; 5 8�"O� G 0 J ? , ;@5 8 2:5 , 5 *N.C, 5 * 6 & 5�A1=S;�83; *RM 5<E:5 ,3* 8/=CE M ; 5 8 J-G ;@= �I5 >�( 8/2:5 0 JL. ? A E ;H JL,+, J 4 ;uL JNG 8/2:5 > � >

=S615IE�8 =C8 & A * 8 , =RA "� J EB;@=S615 , 8 2B5 A * FBF:=CE M � ( � � H B � 'G, ) L % @'42:5 , 5IH B � ' , ) L % c H B � 'G, ) L P ��� � P H B � ' , ) L

H > 8 = A 57;XL " T 5�615 H EB5 � H H F fa 'G, � �K) V F " a 'G, ��� ) V�$ $�$4V F % a ' , � �*) L L�c�H F ma 'G, ) V F "fa 'G, ) V�$ $�$�V F % a ' , ) L L�"

�<N

Page 24: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

��2:=S; A * F:FB=CE M =>;4�5 .>. 615 H E:576m@ GKJL, = G F a ' , ��� ) c � a ' , ��� ) 8/2:5IE F @ � c �&, ��� GKJN, ; J A 5B� Y B @18 2'?B;

F @ � c H �&, � � L , Y 'G, ) @ 8 2'?B; F a ' , ) c � a ' , ) " ��2B5 G * 0 8 8 2 * 8 =C8 =S; * E J E�8 J 2 J A J A JN, F:2:=S; A =>;

J % ?'= J ?B;; J 8 2:5 F , J'JNG 4= .C. % 5 J A =P8 8 5<6 "�� J 490 J EB; =>6:5 ,OB � ' , ) P ��� � P]B � 'G, ) "�$1=CEB0I5�,Q=S;= ,/, 5<61?B0I= % . 5 * EB6

; =CEB0I5 * �?B0 . =S615 * E 6 J A * =>EQ=S; * F , =>EB0 =>F *N. =S615 *N. 6 J A * =>E8@ ' , ) =>; A * A1= A *-. "���2'?B; B � 'G, ) =S; * H 5 . 6 HK42:5<E

& J ? A ? . 8/=CF . & GK,/J A 5 . 5 A 5<E�8/; J-G(B � 'G, ) L . 5 8+?B;�0 *-.>. =C8 D " � J 4!; =CE 0 5 B � 'G, � � ) P � ��� P B � ' , � � ) 2 * ; *

M 5<E:5 ,3* 8/=CE M ;@5I8 JNG ; =��<5 >Q@ V @'; J 6 J 57; D % * ; * E B A J 61? . 5 ; =CE 0 5 � =>; * E J E�8 J 2 J A J A JL, FB2:=>; A " $1?:F:F J ; 5

4�5 0 *-.>. 8 2B=>; M 5IE:5 ,/* 8 =>E M ;@5I8 ��cO. W� 0V $�$�$�VXW % 0 " $1=CEB0I5 � =>; *�M 5IEB5 ,3* 8 =>E M ;@5I8 =P8 M 5IE:5 ,/* 8 57; 6V�$ $�$�V % @-8 2B5

E * 8 ? ,3*-.:M 5IE:5 ,/* 8 =>E M ;@5I8 GKJN,�D % @ JN, =CE J 8/2:5 , 4 JL, 6:; 8/2:5�0 JN. ? A EB; J-G 8/2:5 > � > =>6:5IE�8 =C8 & A * 8 , = A " ��2 * 8 =S; GKJL,

5 * 032( ; 8 2B5 , 5 5�A1=S;�8 \ � ; V $�$�$4V \ % � ; Y B ; ?B032 8/2 * 8 \ � ; W a ��� � a \ % � ; W % c�<;�"��5 A 5 A % 5 , % J 8/2 8 2B5<; 5

5 . 5 A 5<EL83;4@L8 2B5 W ; � ; * EB6 8/2:5� ; � ;�@ *-, 5 *-.>. 0 J ; 5 8/; J-G 'G, ) " $'=>EB0 5"\ � � ; Y B GKJL, ��c V V $ $pV?>�@ V * EB6 = c V V $�$ $�V?> @

8 2'?B;O\ � � ; a 'G, ) Y B � 'G, )�GKJN, �lc V V�$�$pV?>�@ V * EB6 = c V V�$�$ $�V?> "���2:5<EQ=CED8 2B5 D c B � 'G, ) ?L5<0 8 JN, ; F * 0 5 D %

4�5�2 * ?N5�8 2 * 8�Hg\ � ; a ' , ) L W� da���� �0a Hg\ % � ; a 'G, ) L W % c ; * EB6O8 2'?B; 4�5�2 * ?L5 * M 5IE:5 ,/* 8 =>E M ; 5 8 GKJL,OD%

; A *N.C. 5 , 8/2 * EO8 2B5 % * ; =>;�. V�$ $�$�V % 0 @ * 0 J E�8 ,3* 61=S0 8/= J E "���2�? ;�8 2:5 ; A *-.>. 57;�8 M 5<E:5 ,3* 8/=CE M ; 5 8 =S; J-G ;@= �I5 > "

��2:5 * % J ?L5 8 2B5 JN, 5 A =S; ?'=P8 *-.>. & = A F JL, 8 * E�8 GKJN, 8 2B5�F , J'J-G 8 2 * 8�8/2:5 G * 0 8 JN, = � * 8/= J E =S;�?BE:= ��?:5�" T 5�2 * ?L5

J E . &DA JN, 5�8 2:=>E M 8 J ;@2 J 4 % 5 GKJL, 5 4�5 0 * ERF ,/J ?N5 J ? , A * =>ER8 2:5 JN, 5 A " T 5 4= .>. E J 4 ; 2 J 4 8/2 * 8 B � 'G, � ) =>;

=S; J A JL, FB2:=>0�8 J , % B � ' , � � % ) "

���b�����"$ CfP�� � G B =>; * EU=>E�8 5 ML,/*N. 6 J A * =>E * EB6 ,"=S; * E"5 . 5 A 5IE�8 J-G B 8 2:5<EU8/2:5 B A J 61? . 5

B � ' , � ) =>; =S; J A JN, F:2:=S0 8 J 8/2:5 B A J 61? . 5�, % B � ' , � � % ) " *�-658554X2 � 5I8 B % 5 * � ?B0 . =S615 * E 6 J A * =>E * EB6

. 5 8�, % 5 * E 5 . 5 A 5IE�8 JNG,B " � J EB; =>6:5 , 8/2:5DE * 8/? ,/*N. 2 J A J A JN, F:2:=S; A GK, J A B J E�8 J , % B � 'G, � � % ) @42:5 , 5

� H F L,c�, % F a ' , � � % ) " � J E ;@=S615 , 8 2B5�$L5 , E:5 .�J-G 8/2:=>; A * F:FB=CE MR* E 6QE J 8/=>0I5 8/2 * 8 � H F L�c N = G�* EB6 J E . & = G

, � � % � , % F " ��2'?B;�; =CEB0I5 B =>; * ?BE:= ��?:5 G * 0 8 JN, = � * 8/= J E 6 J A * =>E8@ , � � % � , � F = G * E 6 J E . & = G , � � F @1; J 8 2:5�$N5 , E:5 .

J-G � =S;6' , � ) "���2:5 , 5 GKJL, 5 %'& 8/2:5 HB, ;�8 2 J A J A JN, F:2:=S; A 8 2:5 JN, 5 A @'8 2:5 B A J 6:? . 5 B � 'G, � ) =S;�=S; J A JL, FB2:=>0 8 J

8 2B5 B A J 61? . 5 , % B � ' , � � % ) "

T 5 0 * EOE J 4 F ,/J ?N5 8 2 * 88 2:5 G * 0 8 JL, = � * 8 = J EO=>;?BE:= ��?:5�"

���b�����"$ C P�� � 5 8 B % 5 * � ?B0 . =S615 * EQ6 J A * =CE8@ * EB6 . 5 8 , V�$�$ $�V , % V � V $�$�$�V � � % 5 = ,/, 5761?B0 = % . 5

5 . 5 A 5<EL83; JNG`B " ��2B5IE+= G � c B � 'G, � � )KP ��� � PbB � 'G, % � �9) =>; =>; J A JN, F:2:=S0�8 J # c B � ' � � ��)?P � ��� P�B � '� � ��� ) @

� V

Page 25: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

* E 6 = G(B � 'G, ; ��� ) =>; * G * 0 8 JN,�J-G � @�8/2:5IE =C8�=S; J E:5 J-G 8/2:5 G * 0 8 JN, ; JNG #U@ JN, =>E J 8 2:5 , 4 JN, 6:;�8 2:5 , 5 5�A1=>;@8/;

* �D; ?B032R8 2 * 8�,5; c1� � * EB6 � ;dc � � " *�-658554X2 � 5 8 B % 5 * �?B0 . =S615 * E 6 J A * =>E8@ . 5I8�, V�$ $�$�V , % V � V�$�$ $4V � �

% 5 = ,/, 5761?B0I= % . 5 5 . 5 A 5<EL83; J-G�B * EB6 ;@?:FBF J ;@5 8 2 * 8+8 2B5 B A J 61? . 5(� c B � ' , � � )/P ��� � P B � 'G, % � � ) =>;

=S; J A JL, FB2:=>0 8 J 8 2:5 B A J 61? . 5 # c B � ' � � � ) P ��� � PeB � '� � ��� ) "R� J 4 ; ?:F:F J ; 5 8 2 * 8 4�5 ML, J ?:F *-.>. 8 2B5

G * 0 8 JL, ;�42:5 , 5�, ; c�, � 8 JNM 5I8 2:5 , " T 5 4 * EL88 J % 5 * % . 5�8 J 0 J EB; =S615 , J E:5+= , , 5<61? 0 = % . 5 * 8 * 8 = A 5 * E 6D; 2 J 4

8 2 * 8 % J 8/2�; =>6157; 2 * ?N5 8/2 * 8 0 J A A J E G * 0 8 JN, " T =P8/2 J ?:8 .>J ;/; J-G M 5IE:5 ,/*N. =C8 & @ . 5 8 ?B; 0 J EB; =>6:5 , � ? ;�8 J E:5 J-G

8 2B56, ; � ;�@ ; * & , " � 5 8�, c �� ���� ���

, ; ��� @ * E 6 . 5I8��Cc �� ���� ���

� ; � � ""� G 0 J ? , ;@5B,5�9� =S; ;@8 = .C. =>; J A JL, F:2B=>0 8 J

, � # " � & 8/2:5 JL, 5 A ; � " � * EB6 � " J`@ * EB6"; =>EB0 5 B =S; * ?:E:= ��?:5 G * 0 8 JN, = � * 8/= J E"6 J A * =>E8@!, �9� =>;�=S; J A JN, F:2:=S0

8 J�B � ' , � � ) P ����� P B � 'G, ��� )�* EB62,5� # =S;�=S; J A JL, FB2:=>0�8 J�B � ' , � ��) P � ��� P]B � 'G, � � ) 42:5 , 5�54?N5 , & ; * =S6G * 0 8 JL,�J-G , �9� =>; *�G * 0 8 JN,�J-G � * EB6 54?L5 , & G * 0 8 JN,J-G , � # =>; *�G * 0 8 JN,�J-G #U@ * EB6 *-. ; J 5�?N5 , & G * 0 8 JN,�J-G � J-G

8 2B5 GKJN, A B � ' , % ) =>; *�G * 0 8 JN,J-G , �9� * EB6 54?L5 , & G * 0 8 JL,�J-G # J-G 8 2 * 8 GKJL, A =>; *N. ; J *�G * 0 8 JL,�J-G , � # " � , =>E

J 8/2:5 , 4 JN, 6:; 4�5�4=>F 5<6 J ?18 *N.C.1JNG 8/2:5 G * 0 8 JN, ; J-G 8 2:5 GKJL, A B � ' T ) 42:5 , 5 T�c , % 570 * ?B;@5�=>E 8 2 * 80 * ; 5 T � , � "

��2:5 E'? A % 5 ,�JNGG * 0 8 JL, ;�=>EA, �9� * EB6A, � # A ?B;@8 % 5 8/2:5 ; * A 5 %'& 8 2:5 JN, 5 A � " � " T 5 ; 2 *N.C.�*-. ; JD* ; ; ? A 5

8 2 * 8 ^�r ����� r ^* * E 6 �

�r � ��� r � " T 5 4 * E�8�8 J ;@2 J 4!8 2 * 8 � ; c ^ ; " �5 A 5 A % 5 , 4�5 $'E J 4�8 2 * 8

B � ' , � � )MP ����� P B � ' , ��� ) =>;=>; J A JN, F:2:=S0�8 J�B � 'G, � � )#P ��� � P B � ' , � � ) "�$1?:F:F J ; 58 2 * 8 � ; � ^ ; E J 8/=>0I5 J-G0 J ? , ;@5�8 2 * 8 , � � B � ' , � � )9P � ��� P , � � B � 'G, ��� ) =S;�;�8/= .>. =S; J A JL, FB2:=>08 J , � � B � 'G, � � )9P ��� � P , � � B � 'G, � � ) "� ?188 2B5 HB, ;�8 G * 0 8 JL, = � * 8 = J ED0 . 5 *-,/. & 2 * ; * .>*N, M 5 , A =CE:= A ? A M 5IE:5 ,/* 8 =>E M ; 5 8 % 5<0 * ?B; 5 A ? . 8/=CF . & =CE M %�& , � �

4=>F 57; J ?:8 *-.>.BJNG 8/2:5 G * 0 8 JL, ; M 5IEB5 ,3* 8 576 %�& , �� 42:5 , 5 ^ � r ^ ; % 5<0 * ?B; 5 J-G 8/2:5 JL, 5 A � " ��* E 6 8 2:5 G * 0 88 2 * 8

^�r � ��� r ^* * EB6 �

�r � ��� r � " � ?:8 8 2 * 8�=S; * 0 J E�8 ,3* 61=S0 8/= J E " T 5 *-,/, =R?L5 * 88 2:5 ; * A 5 0 J E�8 ,3* 6:=>0 8 = J E

= G 4�5 ; ?:F:F J ; 5�8/2 * 8 ^ ; � � ; " ��2:5 , 5 GKJL, 5 � ; c ^ ; " ��2:=>;�F ,/J ?N57;�8/2 * 8 *-.>. 8 2:5 G * 0 8 JN, ; J-G � *N, 5 G * 0 8 JN, ; J-G # "

� ?18�4�5 0 * E *-. ; J F , J ?L5�8 2 * 8 54?L5 , & G * 0 8 JL, J-G # =S; * G * 0 8 JN, JNG � =>EQ5�A * 0 8 . & 8/2:5 ; * A 5 4 * & "���2'?B;4@

8 2B5 , 5 =S; J E . & J E:5 4 * & 8 J 5�A1F , 5<;/; * H E:=P8/5 . & M 5<E:5 ,3* 8/5<6 B A J 61? . 5 * ; * 61= , 5<0 8+F , J 61? 0 8 J-G 5 . 5 A 5IE�83; J-G

8 2B5 GKJN, A B � P]B � 'G, � � ) P ��� � P]B � 'G, % � �*) "� J 4 ; =CE 0 5Q4�5 *-. ; J 2 * ?L5O8/2:5 JL, 5 A ; � " V * EB6 � " �`@�4�5Q2 * ?N5RF ,/J ?N5<6�8 2 * 8 4�5 0 * E 4 , =P8/5 * H E:=C8 5 . &

M 5<E:5 ,3* 8/5<6 B A J 61? . 5 =>E * ?:E:= �L?B5 GKJN, A "���2:=S; GKJN, A =S; 0 J A F . 5 8 5 . & * E *N.CJLMNJ ?B;�8 J 42 * 8 4�5+2 * ?N5 *-.>, 5 * 6 &

6 J E:5 =CE H E:=C8 5 � % 5 . = * E ML, J ?:FB;�" � J 8/=>0I5 8/2 * 8 ; =>EB0 5 *OH E:=C8 5 . & M 5<E:5 ,3* 8/5<6 � A J 61? . 5 =>; *N. ; JR* E � % 5 . = * E

� �

Page 26: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

MN,/J ?BF 4�5 2 * ?L5 F ,/J ?N576R8 2 * 8+54?L5 , & H E:=P8/5 . & M 5<E:5 ,3* 8/5<6 � % 5 . = * E MN,/J ?BFU=S; =S; J A JN, F:2:=S0+8 J 6:= , 5<0 8�F ,/J 61?B0 8

� % P ������ P � ��� P �

� ��� @�42:5 , 52, ; =S; F , = A 5K@ * E 6 8 2:=S; =>; 8 2:5 J E . & ;@?B032�F , J 61? 0 8 8/2 * 8 8 2:5 H E:=C8 5 . &

M 5<E:5 ,3* 8/5<6R� % 5 . = * E ML, J ?:FR=>;�=>; J A JN, F:2:=S0 8 J " ��2:=>; 0 J A F . 5I8 5<; 8 2:5 F , J'JNG JNG�J ? , A * =CER8/2:5 JL, 5 A " T 5 4= .C.

E J 4 MLJ J E 8 J 0 J E ;@=S615 , ; J A 5+;@F 5<0I= *N. 0 * ; 5<; J-G � ?B0 . =S615 * EO6 J A * =>EB;�@�F *-, 8 =S0 ? .S*-,/. & F JN. & E J A = *-. ;4@'42:5 , 5�4�5

0 * E * F:F . & J ? ,�, 57;@? . 8/;�"

��

Page 27: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

*,+.-0/ �82 ��3e� � ���Q=>;K+`/0=Q5 3 /05U*,5 � � 3 5 �]=Q+�� �

� 5 8 D % 5 * H 5 . 6 " � 5 A 5 A % 5 , 8 2 * 8 D�E F.G =>; * � ?B0 . =S615 * E 6 J A * =>EU4=C8 2Uo�H� H F L Lbc&oM��i\* � H F L GKJL,

* E & � H F L Y D�E F.G " � J E ;@=S615 , 8/2:5 � % 5 . = * E MN,/J ?:F D � @�42B5 , 5 � =>; * E * 8 ? ,3*-. E'? A % 5 , " � JN, A *N.C. & 4�5

A ? . 8/=CF . & 5 . 5 A 5<E�8/; J-G(D � %�& 5 . 5 A 5IE�8/; J-G D 8 JDM =p?N5 ?B; * E � 61= A 5IEB; = J E *-. ?N570 8 JN, ;@F * 0 5�" � J 4�54?N5 , @ 4�5

0 * E *N. ; J A ? . 8 =>F . &O%'& F JN. & E J A = *-. ; GK, J A D�E F.G ; J .CJ E M * ;4�5+0 *N, 5 G ? .C. & 615 H E:5+42 * 8 4�5 A 5 * E " � 5 8 � % 5

* EN� �Q� A * 8 , =RA 4=C8 2 5IE�8 , =>5<; GK,/J A 8 2:5 H 5 . 6 D " $'?:FBF J ;@5�8/2 * 8�� H F Ldc T#% F % a ��� � a�T�� 8/2:5IE . 5 8 ?B;615 H EB5

� H � L"c T&% � % a T&% �% a ��� �Ma T�� �R42:5 , 5 �R=>;�8/2:5 =>6:5IE�8 =C8 & A * 8 , =RA " � J 8 =S0 5 8 2 * 8 GKJN, F JN. & E J A = *-. ;

� V � Y D�E F.G @dH� a �.L�H��(L,c � H � L a �mH � L * E 6]H� �.L�H��(LOc � H � L �mH �(L�@ ;@=>EB0 5 42:5<E * 6B61=CE M JN, A ? . 8/=CF . & =CE M

F JN. & E J A = *N. ;�=>E F 4�5 0 JL.C. 5<0 8 . = $N5 8 5 , A ; * EB6"4�5 0 * E"F 5 ,@GKJL, A 8/2:5 ; * A 5 F , J 0 5761? , 5 J E"F JN. & E J A = *-. ; J-G

A * 8 , =S0 57;�" � 5 8?B; 615 H E:5+2 J 4 A ? . 8 =>F . =>0 * 8/= J E GK,/J A D�E FqG 4 JN, $';�" � * $N5 * E & ��Y D � * EB6 * E & � H F L Y D�E F.G

* E 6 . 5 8 ?B; 615 H E:5 � H F L � c � H��(L � "Q$'=>EB0 5 H � a �`L4H �(L c � H � L a �mH �(L * EB6eH� �.L�H��(L c � H �(L � H��(L�@ 8/2:=>;

; * 8 =S; H 5<;�8/2:5+F ,/J F 5 , 8 =>5<;�E:5I57615<6O8 J % 5 * A J 61? . 5 "

# ��� 3 =g/0=Q5 3 �8PQ1 T 5615<E J 8/5 8 2:5 A J 61? . 5 D � 4=C8 2 ;/0 *N.>*N, A ? . 8 =>F . =>0 * 8 = J E GK,/J A D�E FqG 42:5 , 5 � H F L � c

� H � L � GKJL, � H F L Y D�E F.G @ �Y D � @ * EB6R; J A 5 H A1576 � � � A * 8 , =RA �O@ * ;bH D � L�� "

� 5 8�?B; , 5 A =>EB6 J ? , ;@5 . ?L5<; 8 2 * 8 8�4 J � ��� A * 8 , =>0I5<;�@ � * EB6 � " @ *N, 5 0 *-.>. 5<6U; = A = .S*-, = G 8 2:5 , 5 5�A1=>;@8/;

; J A 5�=CEi?N5 , 8/= % . 5 A * 8 , =RA�� ;@? 032O8/2 * 8�� � � c � " "�� . ; J , 5<0 *N.C. 8/2 * 8�;@= A = .S*-, A * 8 , =S0 57;�2 * ?N5�8/2:5 ; * A 5

032 *N,/* 0 8 5 , =S;@8 =S0�F JN. & E J A = *-. * EB6O8 2'?B;8/2:5 ; * A 5+5I= M 5<Ei? *N. ?B5<;bH � * & KN W L " T 5 ; * & 8 2 * 8 * A * 8 , = AD=>; 61= *-MNJ

E *N. = � * % . 5 = G =C8 =S;�;@= A = .S*-, 8 JO* 61= *NMNJ E *-. A * 8 , = A "�� J 8/=>0I5 8 2 * 8 = G � c� � " � 8 2B5IE � � c� � " � � "

��2:=S; =S; 5 * ; & ; =CE 0 5 � � c H�� � " � L4H�� � " � L�H�� � " � L � ��� H�� � " � Lc � � " H�� � L � " H�� � L

��� � H�� � L � " � c�� � �" � "

� � � � +m2 � G � c�� � " � GKJL, A * 8 , =S0 5<; � V � " @ * EB6�� H�� =CEi?N5 , 8/= % . 5 Lt@-8/2:5IE GKJL, * E & � H F L Y D�E FqG @

� H � Ldc��6� H � " L�� " ��2:=>; =>; 5 * ; & 8 J ;@5<5K@ GKJL, = G � H F L�c T�F � a ��� �>aT XF a�T � @N8 2:5<EN� H�� Ldc T

�H � L � a

��� � a�T! H � L a�T�� �Ic T�H�� � " � L � a���� � abT& H�� � " � L abT�� H�� ��� Ldc�� T

�� " � � a�� ��� a � T�� ��� c

��H� H�� " L L�� @:42B=>032Q0 J A F . 5I8 57;�8 2:5+F ,/J'J-G "

T 5 4= .>. E J 4 F , J ?L5 * EU=CE�8 5 , 57;�8/=CE M 8 2B5 JN, 5 A , 5 .S* 8/=CE M 8 2:57;@5 A J 61? . 5<; 8 J 8 2:5 F ,/J % . 5 A J-G ;@= A = .S*-, =P8 &

� �

Page 28: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

J-G A * 8 , =S0 57;�"

���b�����"$ �8PQ1 $'?BF:F J ;@5 8 2 * 8 � * EB6 � " *N, 5I� � � A * 8 , =S0 57; 4=P8/2 5IE�8 , =>5<; GK,/J A * H 5 . 6

D "Q��2:5IEhH D � L���=>; =>; J A JN, F:2:=S0 8 J H D � L��

�= G * E 6 J E . & = G � * EB6 � " *-, 5 ; = A = .S*-, " *�-658554X2 $1?:F:F J ; 5

8 2 * 8 � * EB6 � " *N, 5D;@= A = .S*-, A * 8 , =S0 57;4@ 8/2 * 8 A 5 * EB; 8/2 * 8 � � � c � " GKJL, ; J A 5 =>Ei?N5 , 8/= % . 5 A * 8 , = A

� "U� 5 H EB5 � ( H D � L ��� H D � L �

�%'& � H � L�c � � GKJN, * E & � Y D � " ��2:=S; A * FBF:=CE M 0 . 5 *N, . & ; * 8/=>; H 57;

8 2B5 * 6B61=P8/=R?L5 2 J A J A JN, F:2:=S; A F ,/J F 5 , 8 =>5<;�@ * EB6 ; =>EB0 5 � =>; =CEi?L5 , 8 = % . 5M@ =P8 =>; * % = � 570 8 = J E * ; 4�5 .C. ""� .C. =>;

8 2 * 8 =S; . 5 G 8 8 J ;@2 J 4 =S;+8/2 * 8 � H � H F L � LIc � H F L � H � L ""� JL, * E & � Y H D � L���* EB6 � H F L Y D�E FqG 4�5 2 * ?L5

8 2 * 8 � H� H F L � Lbc � H � H � L � L�c � H��6� H�� " L�� � L�c � H � " L�H�� � L�c � H F L � H � Lt@ 42:=>032 0 J A F . 5I8 5<;

8 2B5 HB, ;�8�61= , 570 8 = J E "�� J 4 8 J F ,/J ?N5�8/2:5 0 J Ei?N5 , ; 5K@ ;@?:FBF J ;@5+8/2 * 8�H D � L���* EB6 H D � L��

�*N, 5 =>; J A JN, F:2:=S0 * ;

B A J 61? . 5<;�"�� 5 8 � % 5 * E & =S; J A JL, FB2:=>; A % 5I8�4�5<5IEQ8/2:5 A @ �5 A 5 A % 5 , 8/2 * 8 � =>; 0 J A F . 5I8 5 . & 615 8/5 , A =CE:576

%'& =P83; * 0 8/= J E J E * M 5IE:5 ,/* 8 =>E M ; 5 8 " ��2:5 A J ;�8 J % ?'= J ?B; M 5IEB5 ,3* 8 =>E M ;@5I8�=S;�.< V�$ $�$�V �0 @�8 2B5 0 JN. ? A EB; J-G

8 2B5 � �Q�D=>6:5IE�8 =C8 & A * 8 , =RA " � J 4 � A ?B;@8 *-. ; J A * F 8/2:5<; 55 . 5 A 5IE�83; 8 J *�M 5<E:5 ,3* 8/=CE M ; 5 80@N; * & . T V $�$ $�V T�0 "

$'=>EB0 5N. V�$ $�$4V �0 0 * E M 5<E:5 ,3* 8/5 *N.C. 5 . 5 A 5IE�8/; JNG H D � L �

�?B; =>E M J E . & F JN. & E J A = *N. ; JNG 615 MN, 5I5bN HK42:=S032R=>;

� ?B;@8 A ? . 8/=CF . =S0 * 8 = J E %'& 5 . 5 A 5IE�83; J-G � L�@ ; J 0 * E . T! 0V $�$ $4V T�0 "���2'?B; � =S; * A * 8 , = AD8 ,3* E ; GKJL, A * 8/= J ED8 2 * 8

8 * $N57; � 8 J Z � @B42:5 , 5 8 2B5 0 JN. ? A EB; JNG Z *-, 5 T! 8/2 ,/J ? M 2 T�" � J 8 =S0 5�8/2 * 8 ; =CEB0I5�8 2:5 0 JN. ? A EB; J-G Z ;@F * E

D � * ; * E D A J 61? . 5K@ Z =S; =>Ei?N5 , 8 = % . 5 "�� J EB; =>6:5 , 8 2:5�=CEi?L5 , ;@5 =S; J A JN, F:2:=S; A �� @-;@=>EB0I5=>E ?L5 , ;@5 A * 8 , =>0I5<; *-, 5

?:E:= ��?:5K@ �� H � L,c Z � "+��2:5 , 5 GKJL, 5 GKJN,�= c V V�$ $�$�V � @ Z � ; c � H�� ; L,c � H F ; L"c F � H ; L,c � "4Z ; "

��2'?B;�@:;@=>EB0 5 Z � <;fc � " Z ; GKJN, *N.C. <; @:4�5 $'E J 4 Z � c � " Z @ * EB6 8 2:5 , 5 GKJN, 5 Z � Z c � " ; J � * EB6

� "�*N, 5+; = A = .>*N, @142:=S032R0 J A F . 5 8/5<;�8/2:5+F ,/J'J-G "

T 5 0 * E ?B; 5R8 2B5R8 2:5 JN, 5 A ; 4�5Q615�?N5 .CJ F 576�=>E ;@570 8 = J E ; * EB6 � 8 J 5�A * A =>E:5D8 2:5QE * 8 ? , 5 J-G 8 2B5<; 5

A J 61? . 5<; GKJL, F *-, 8 =S0 ? .S*-, A * 8 , =S0 57;�" T 5+4= .>. HB, ;@80 J EB; =S615 , 42 * 82 * F:F 5IE ;42:5IE � =S; * 61= *NMNJ E *-. A * 8 , =RA "

���b�����"$ �mP�� � G D =S; *OH 5 . 6 * EB6�� =S; * 61= *NMNJ E *-. A * 8 , =RAU4=C8 2 5IE�8 , =>5<; GK,/J A D 8 2B5IE 8 2B5

D�E F.G A J 6:? . 5 D � 42:5 , 5 � H F L � c � H �(L � GKJL, � H F L Y D�E F.G* EB6 � Y D � @�=>; =S; J A JL, FB2:=>0 8 JSD�E F.G� ' F @

W �)DP ����� P D�E F.G� ' F @hW�) 42:5 , 5 W V�$ $�$�VuW

�*-, 5 8/2:5D5IE�8 , =C57; *N.CJ E M 8/2:5Q61= *NMNJ E *-.�J-G � " *�-658554X2 � 8 =>;

; ? � 0 =>5IE�8�8 J F ,/J ?N5+8/2 * 8 8 2:5�$N5 , E:5 . J-G 8/2:5 E * 8 ? ,3*-. ;@? , � 570 8 =p?N5 2 J A J A JL, F:2B=>; A � 8/2 * 8 8 * $L5<;�H D�E F.G L � 8 J

8 2B5 D�E F.G A J 61? . 5_H D � L�� ? ;@=>E M 8/2:5 M 5IE:5 ,/* 8 =>E M ;@5I8(.< 6V�$�$ $�V �0 @ HK8 2:5D0 JL. ? A EB; J-G 8/2:5A� � � =S615IE�8/=P8 &

�KJ

Page 29: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

A * 8 , =RAqLt@N=S;�' F @�W� )�P ��� � P ' F @�W�) "�� J 4 � H H� H F L V $�$�$�V �

�H F L L L(c � H F L �a ��� � a �

�H F L

�" � J 8 =S0 5 8 2 * 8

; =CEB0I5 � =S; 61= *-MLJ E *N. � % =S; 61= *-MLJ E *N. 4=C8 2 5<EL8 , =>5<; *N.CJ E M 8 2:5 61= *-MLJ E *N. J-G W % V $�$�$�VXW�% " ��2'?B;�= G � H F L�c

T&% F % alT&% F% a ��� � alT! F alT�� @-8 2:5<EN� H��(L c T&% � % a � ���galT! � a�T�� ��c ��� � H W� L N $ $�$ N

N � H W " L $ $�$ N$�$�$ $�$�$ $ $�$ $ $�$N N $ $�$ � H W

�L

��"

$'?:FBF J ;@5 8/2 * 8 � H H � H F L V�$ $�$4V ��H F L L L c N.@ 8 2:5<E � H � L a ��� �Ma �

�H��(L

�c N " ��2:5 , 5 GKJN, 5N� H W Lbc N.@

� " H W " L!c%N * EB6 ; J J E *N.C. 8 2:5 4 * & 8/2 , J ? M 2���H W�L!c N " ��2'?B; F @ W ; 6:=R?'=S615<;A�9; H F Lt@ ; J 8 2:5 , 5 GKJN, 5

� 0\ � H ' F @ W �) V�$ $�$�V ' F @ W�) L�" � ?18�0 . 5 *-,/. & GK,/J A 8 2:5 * % J ?L5 .CJLM =S0<@fH ' F @UW ) V $�$ $4V ' F @ W

�) L � �\ �

* ;�4�5 .>. @142:=>032Q0 J A F . 5 8 57;�8 2:5+F , J'J-G "

� J 8 =S0 5�8 2 * 88/2:=>; , 5<; ? . 8�5�A'8 5<EB6:;�8 J *N.C. 61= *NMNJ E *-. =�� * % . 5 A * 8 , =S0 57; * ;4�5 .C. @';@=>EB0 5�= G ��=>;�61= *NMNJ E *-. =�� * % . 5

=C8O=S;O; = A = .>*N, 8 J�* 61= *-MLJ E *N. A * 8 , =RA nC@ * EB6 8 2'?B;_H D � L�� =S;O=>; J A JN, F:2:=S0D8 J H D � L�� "���2:5<E ;@=>EB0I5 8 2B5

5I= M 5<Ei? *N. ?B5<; J-G n *-, 5�; = A F . & 8/2:5�5IE�8 , =C57; *-.>J E M 8 2:5+61= *-MLJ E *N. * EB6O; =>EB0 5�;@= A = .S*-, A * 8 , =S0 57;2 * ?N5 8 2:5+; * A 5

5I= M 5<Ei? *N. ?B5<;�@ H D � L�� =S; =S; J A JL, FB2:=>0 8 J D�E F.G� ' F @ W� ) P � ��� P D�E FqG � ' F @eW�) @ 42:5 , 5 W V�$ $�$�VuW

�*-, 5 8 2B5

5I= M 5<Ei? *N. ?B5<; J-G � " � J 8/=>0I5�8/2 * 8�8 2B5 F ,/J 61?B0 8 JNG 8 2:5�H F @CT ; L� ;�=S;�8 2:5+032 *N,/* 0 8/5 , =>;@8 =S0F JN. & E J A = *N.BGKJL, � " � E

.S* 8 5 , 8 2:5 JN, 5 A ;�@14�5�4= .C. 54A'8 5IE 6 8/2:=S; , 57;@? . 8�8 J E J E! 61= *NMNJ E *-. =�� * % . 5 A * 8 , =>0I5<;�"

$'?BF:F J ;@5 4�5 *N, 5 M =R?L5IE * F *-, 8 =S0 ? .S*-, F JN. & E J A = *-. ,:H F L * E 6Q4�5+4 * E�8 8 J 0 J EB;@8 , ?B0 8 * E D�E FqG A J 61? . 5K@

H D � L � =>; J A JL, F:2B=>0�8 J�D�E F.G� ,:H F L " T 2 * 8 A * 8 , = AQ;@2 J ? . 6R4�5 ?B; 5�� T 5 4= .>. H , ;@8�; 2 J 4 2 J 498 J 0 J E ;�8 , ?B0 8

* A * 8 , =RA 4=P8/2 * M =p?N5IED032 *-,3* 0 8 5 , =S;�8/=>0 F JN. & E J A = *N. "

���b�����"$ �mP � ��2:5�8 2:5 032 *N,/* 0 8/5 , =>;@8 =S0�F JN. & E J A = *-. JNG 8 2B5 A * 8 , = A

Z c�������N N N $ $�$ N @OT��V N N $ $�$ N @OT! N V N $ $�$ N @OT#"N N V $ $�$ N @OT �$�$�$ $ $�$ $�$ $ $ $�$ $�$ $ $�$�$N N N $ $�$ V @OT

������

=S;�,:H F L"c H @ V L � H�� � a T� � �

aL��� �Ka T � a T � L�" *�- 5f5 4X2 T 5 4= .>. ?B; 5 =>EB61?B0 8 = J E J E ��" � ? , % * ; 5

0 * ;@5 =S;Q� c � " � J 8/5 8/2 * 8�8/2:5 615 8/5 , A =>E * EL8 JNG 8 2:5 A * 8 , = A � @ � @OT �V @OT @ � � =>;�� � aeT � a T � " � J 4

* ;/; ? A 5 8/2 * 88/2:5 ;@8 * 8/5 A 5IE�8=S;�8 , ?:5�?:FD8 J � @ V @ * EB6D0 J EB; =>615 , 8 2:5B� � � A * 8 , =RA Z9A =CE'?B;8 2:5+=S615IE�8/=P8 &

� �

Page 30: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

A * 8 , =RA A ? . 8/=CF . =>5<6 %'& � "

Z�@ � ��c�����@ � N $ $�$ N @OT��V @ � $ $�$ N @OT N V $ $�$ N @OT "$ $�$ $ $�$ $ $�$ $�$ $ $�$�$N N $ $�$ V @OT

� @ �

���� $

T 5 4= .>. ?B;@5O0 J-G * 0 8 JN, 5�A1F * EB; = J E *-.>J E M 8 2B5 HB, ;�8 , J 4�8 J 0 *-. 0 ? .S* 8/5 8 2:5D615 8/5 , A =CE * E�8�"Q� J 8 =S0 5 8/2:5 J E . &

0 JL. ? A E ;O4�5UEB5I5<6 8 J 0 J EB0 5 , E J ? , ;@5 . ?L5<; 4=C8 2 *-, 5 8/2:5 H , ;@8 * EB6 8/2:5 .>* ;�8 " ��2'?B; J ? , 032 *-,3* 0 8/5 , =>;@8 =S0

F JN. & E J A = *N. =>;

@ �so� ������@ � N $�$�$ N @OT& V @ � $�$�$ N @OTM"N V $�$�$ N @OT �$�$�$ $ $�$ $�$�$ $ $�$ $�$ $N N $�$�$ V @OT

� @ �

���� a H @ V L � H @OT�� L o� � ��� V @ � N $�$ $ N

N V @ � $�$ $ N$�$ $ $�$ $ $�$ $ $�$ $ $ $�$N N N $�$ $ V

��$

� & 8/2:5=CE 61?B0 8/= J E8@N2 & F J 8 2B5<; =>; 8/2:5 615I8 5 , A =CE * E�8 J-G 8 2:5 HB, ;@8 A * 8 , =RA =S;OH @ V L � H�� � a�T� � �

" a ��� � a

T " � aCT Lt@ * EB6 0 . 5 *-,/. & 8 2:5 615 8/5 , A =>E * EL8 J-G 8 2:5�;@570 J EB6 A * 8 , = A =S; V " ��2'?B;�8/2:5 032 *-,3* 0 8/5 , =>;@8 =S0�F JL. & E J A = *N.

J-G Z =S; @ �fH @ V L � H�� � a T� � �

" a�� ���0a T " � a T L a H @ V L � H @OT � L�c�H @ V L � H � � a]T� � �

a

��� �0aUT#" � " a T! ��L a H @ V L � T�� c H @ V L � H�� � aUT� � �

a���� �0a T#" � " a]T! � a]T�� L�42:=S032R0 J A F . 5 8/5<;�8 2B5

F ,/J'J-G "

T 5�4= .>. 615<E J 8/5 8/2:5 A * 8 , =RA 0 , 5 * 8 5<6�=>E 8/2:5 * % J ?L5�8 2B5 JN, 5 A GKJL, 8 2B5�; F 570 = H 0 F JN. & E J A = *-. , H F L * ; Z� � ��� "

$'?:FBF J ;@5 ? ;@=>E M 8 2B5+8 2:5 JN, 5 A ; GK,/J A 8/2:5 F , 54?'= J ?B;�;@570 8 = J E ; GKJN,�* F *-, 8 =S0 ? .S*-, A * 8 , =RA � 4�5+5IE 6R?:FR4=P8/2 *

A J 61? . 5�=S; J A JL, FB2:=>08 J�D�E F.G� ' , H F L ) 42:5 , 5 ,:H F L =S;E J 8E:570 57; ; *-, = . & = ,/, 5761?B0I= % . 5 " T 5 0 * E * ;/; ? A 5 4=P8/2 J ?:8

.>J ;/; J-G�M 5<E:5 ,3*-. =C8 & 8 2 * 8�,:H F L=S; J-G 8/2:5 GKJL, A H @ V L � H F � a T� F �

a �����6a T�� L�" � & 8 2:5 F , 5�?'= J ?B;�8 2B5 , 5

5�A1=S;�83; * A * 8 , =RA Z� � ��� 42 J ; 5 032 *N,/* 0 8/5 , =>;@8 =S0F JN. & E J A = *N. =S; , H F L�" T 5�4= .>. E J 4 ;@2 J 4 8/2 * 8 D�E F.G� 'G,:H F L ) =>;

=S; J A JL, FB2:=>0�8 J 8/2:5 D�E F.G A J 61? . 5H D � L������� "

���b�����"$ �mP�C $1?:F:F J ; 5 8/2 * 8�,:H F L Y D�E F.G�* EB6U8 2:5 615 MN, 5I5 J-G , H F L�=>;Q� @ 8 2:5<E D�E F.G� ' , H F L )

=S;+=>; J A JL, F:2B=>0 8 J H D � L�������� " *�-658554X2UT 5 0 * E * ; ; ? A 5 4=C8 2 J ?18 .>J ;/; J-G�M 5IEB5 ,3*-. =P8 & 8/2 * 8�, H F L�=S; J-G 8 2B5

GKJN, A H @ V L � H F � a T� tF �

a ��� �Ma T � L % 5<0 * ?B; 5�'G,:H F L ) c '�`H F L )�GKJL, ; J A 5 A J E:=>0 F JN. & E J A = *-. @ �`H F L�"

$'?:FBF J ;@5 8 2 * 8 , H F L�c H @ V L � H F � a_T� tF �

a � ����a T uF a_T � L�" � J 8/=>0I5�8/2 * 8Q. V V F V F " V�$ $�$�V F � 0�GKJN, A ;

* M 5<E:5 ,3* 8/=CE M ; 5 8 J-G ;@= �I5I� GKJL,D�E FqG � 'G,:H F L ) * ; * E D A J 6:? . 5 * EB6 8/2'?B; * ; * E D�E F.G A J 61? . 5 * ; 4�5 .>. "

� �

Page 31: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

� . ; J E J 8/5 8 2 * 8(. 6V�$ $�$�V �0 GKJN, A ; *RM 5<E:5 ,3* 8/=CE M ; 5 8 GKJN,�D � * ; * E D A J 61? . 5K@ * EB6 8/2'?B; *NML* =>E * ; * E

D�E F.G A J 6:? . 5�" � J 8 =S0 5 8/2 * 8 GKJN,�* E & � H F L a ' , H F L ) Y D�E F.G� ' , H F L ) @ � H F L a 'G, H F L ) c H W� uF �

aL�����<a

W tF a W � L a ' , H F L ) GKJN, ; J A 5 W� V�$ $�$�VuW � Y D " � 5 H E:5 * A * F:FB=CE M$� ( D � � D�E F.G� ' , H F L ) @ 42:5 , 5 GKJL,

* E & T Y D � @ � H T L�c � H T a�� ��� a T��Llc%H T

�F � a1� ���ia T "�F a T L a 'G,:H F L ) 42:5 , 5 T ; =>; 8 2B5

= � 8/2"5<E�8 , & J-G T " � . 5 *-,/. & 8 2:=S; A * F:FB=CE M =S;�4�5 .C. 6:5 H E:576"; =>EB0 5 8 2B5 , 5 =S;�E JQ* A % = M ?:=C8 & =CE 8/2:5 6 J A * =>E J-G

8 2B5 G ?BEB0 8/= J E " ��2B=>; A * FBF:=CE M =S; *-. ; J ;@? , � 570 8 =p?N5R; =CEB0I5R54?N5 , & 5 . 5 A 5<E�8 =>E D�E FqG � 'G, H F L ) 0 * E % 5D4 , =C8@8 5<E

=>E 8 2:5 GKJN, AkT� F �

a ��� �qa T! F a T��ba 'G, H F L ) "���2B=>; A * F:F:=>E M =S; * E * 6:61=C8 =p?N5U2 J A J A JL, F:2B=>; A @

; =CEB0I5 � H � a�� L c � H H �� a��� L �a � ���sa H ��a��

�L �LSc H H �

�a��

�L F � a �����sa H �� �a��� L L a

'G,:H F L ) c H H ��F � aL� ���Ka � L a 'G,:H F L ) L a H H �

�F � aL��� �6a�� L a ' , H F L ) L"c � H � L a � H � L " T 5 *N. ; J

A ? ;�8O; 2 J 4 8 2 * 8 8 2:=S; A * F:F:=>E M =>; * E =CE � 5<0 8/= J E " � JN, 0 J Ei?L5IE:=>5IEB0I5K@ GKJN, * E & � Y D � 4�5Q4= .>. 615<E J 8/5

���iH F L c ��F � a1�����Ka � 42:5 , 5 � V $�$ $�V �

�*-, 5 8 2:5 5<E�8 , =C57; J-G � "D� J 4�; ?:F:F J ; 5 8 2 * 8 � H � L�c � H � Lt@

8 2B5IE � � H F L a ' , H F L ) cO��� H F L a ' , H F L ) @ ; J � � H F L @ ��� H F L Y ' , H F L ) "�$'=>EB0 5 % J 8 2 � � H F L * EB6 ��� H F L2 * ?L5

615 MN, 5I5 . 5<;/;8/2 * E � @�; J 6 J 5<; 8 2:5<= , 61=� 5 , 5<EB0 5�"���2'?B;�;@=>EB0 5 8 2:5 615 ML, 5<5 J-G ,:H F L�=S;�� @-� � H F L @ ��� H F LOc N.@

; J � � H F L c ���(H F L * EB6 � c � " ��2'?B; 8/2:5 A * F:FB=CE M =S; J EB5 8 J J EB5 " � J 0 J A F . 5I8 5R8/2:5QF ,/J�JNG 8/2 * 8 �

=S; * E =S; J A JN, F:2:=S; A =C8 =S; 5IE J ? M 2 8 J ; 2 J 4 8/2 * 8 � H F ; L!c F � H ; L * EB6 8/2 * 8 � H � ; L c � � H ; L GKJN, * E &

��Y D @ % 5<0 * ?B; 5O4�5O0 * E 0 J EB;@8 , ?B0 8 * E & F JN. & E J A = *N.�J ?18 J-G,F � ; * E 6 5 . 5 A 5<EL83; J-G"D " � JN, 5�A * A F . 5K@ = G

8 2B5<; 5 0 J EB61=C8 = J EB;�2 JN. 6m@�E J 8 5+8/2 * 8 � H H F " a F L � LOc � H F " � L a � H F � L"c F � H F � L a F � H � LOc H F " a F L � H � L $

� J 8 =S0 5Q8/2 * 8 � H F ; L c � H Z� � ��� ; L�c � H Z ; L�@42:5 , 5 Z ; =>; 8 2:5 = � 8 2 0 JN. ? A E JNG Z

� � ��� " $'?BF:F J ;@5 8 2 * 8

= Y . V V�$ $�$�V � @ V 0 @L8 2:5<ED0 . 5 *-,/. & @ � H Z ; Ldc � H ; � Ldc F ; a ' , H F L ) c F H F ; a 'G,:H F L ) L(c F � H ; L "�� J 4 GKJL,

= c � @ � H F ; L"c � H F �L"c � H Z

� � ��� � L"c H @OT � F � a @OT

� " F � " a ��� �Ka @OT! F a @OT�� L a 'G,:H F L ) "

�5 A 5 A % 5 , 8 2 * 8 ,:H F L(c F � a�T� F �

a(����� a T! F a T�� @<8/2'?B; F � a ' , H F L ) c H @OT� F �

a @OT� " F �

" a

��� � a8@OT uF a @OT � L a ' , H F L ) @:; J E J 4 4�5 2 * ?L5�8 2 * 8 � H F �L�c F � a 'G,:H F L ) c F H F � a 'G,:H F L ) L c F � H

�L�"

��2:5 , 5 GKJN, 5 � H F ; L c F � H ; L " � ?18 *-. ; J 0 . 5 *-,/. & � H � ; L c � F ; a 'G,:H F L ) c � H F ; a 'G,:H F L ) LIc � � H<; L�"

��2:5 , 5 GKJN, 5 � =>; * E D�E F.G A J 61? . 5+=S; J A JN, F:2:=S; A @142:=S032D0 J A F . 5 8/5<;�8/2:5+F ,/J�JNG "

T 5 4= .>. E J 4 ;@2 J 4�2 J 4!8 J F * ;@8 5 8 2:57;@5 A * 8 , =>0I5<; 8 JLM 5 8/2:5 , 8 JRM 5 8 *N, % =C8 ,3*-, & D�E FqG A J 61? . 5<; J-G 8 2B5

GKJN, A D�E F.G� ' , H F L ) P ����� P D�E F.G� ' , % H F L ) "

� �

Page 32: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

���b�����"$ �8P�� � G�D =S; *"H 5 . 6 8 2:5<E 8/2:5 D�E F.G A J 61? . 5 J-G 8 2:5 GKJL, A D�E FqG � 'G, H F L ) P ����� P

D�E F.G� 'G, % H F L ) =S;�=S; J A JL, FB2:=>0 8 J 8/2:5 D�E F.G A J 61? . 5 D � M 5IE:5 ,/* 8 576 %'& 8/2:5B� � � % .CJ 0 $ A * 8 , = A

Z c ��� Z ��� � ��� N $ $�$ NN Z

� � � ��� $ $�$ N$�$ $ $�$ $ $ $�$ $�$ $N N $ $�$ Z

� � � ���

��V

42:5 , 5B� =S;8 2B5 ;@? A JNG 8/2:5 615 MN, 5I57; J-G 8 2B5 F JL. & E J A = *N. ; , H F L�8/2 ,/J ? M 2�, % H F L * E 6 Z� � � ��� =>;8 2:5 A * 8 , = A

GK,/J A 8/2:5 JL, 5 A J!" U4=C8 2 032 *N,/* 0 8/5 , =>;@8 =S0 F JN. & E J A = *-. , ;XH F L�" *�-658554X2 � = , ;�8 . 5 8 ?B; 615 H EB5 oM;�8 J % 5R8 2B5

615 MN, 5I5 J-G 8 2:5QF JN. & E J A = *-. , ; H F L�" � J F , J ?L5 8/2:=S; 8 2:5 JN, 5 A @4�5R4= .>. ?B; 5R=CE 61?B0 8/= J E J E > " �5<; ? . 8�J!" �

F ,/J ?'=>6157; * % * ;@5O0 * ; 5K@�; J ; ?:F:F J ; 5 8 2 * 8 8/2:5O;@8 * 8 5 A 5<EL8 =>; 8 , ?:5 ?:F 8 J > @ V ""� J E ;@=S615 ,ID�E FqG A J 61? . 5<;

D�E F.G� 'G, H F L ) P � ��� P D�E FqG � 'G, % H F L ) * EB6 H D � L � @bH 42:5 , 5 Z =>;O615 H E:5<6 * ; * % J ?N50L " � & 8 2B5 =CE 61?B0 8/= J E

2 & F J 8 2:57;@=S; D�E FqG � 'G, H F L ) P � ��� P]D�E F.G� 'G, % H F L ) =S;=S; J A JN, F:2:=S0 8 J 8 2:5 D�E F.G A J 61? . 5 D � �� � @ M 5<E:5 ,3* 8/5<6

%'& 8 2B5 % .CJ 0 $ A * 8 , = A

Z � c ��� Z � � � ��� N $ $�$ NN Z

� � � ��� $ $�$ N$�$ $ $�$ $ $ $�$ $�$�$N N $ $�$ Z

� ��� � � ���

��$

$ J 8/2:5 , 5 54A'=S;@8/; * E D�E FqG A J 61? . 5 =S; J A JN, F:2:=S; A @ 0 *N.C. =P8 ��8 2 * 8 8 * $N57;H D � �� � L ��� J E�8 J D�E F.G� ' , H F L )DP

��� � PeD�E F.G� ' , % H F L ) "U� . ; J %'& 8/2:5 JL, 5 A J " �s@ 8/2:5 , 5 54A'=S;@8/; * E =>; J A JL, F:2B=>; A � 8/2 * 8 8 * $L5<;H D�� � L �������

J E�8 J D�E FqG � 'G, % H F L ) " � J 4�8/2:5IE 4�54 * E�8�8 J ; 2 J 4 8 2 * 8"H D � L��U=S; =S; J A JN, F:2:=S08 J H D � �� � L���� P H D�� � L����� � "

� J ; 5I5 8 2:=S;�8 * $N5 * E & �Y H D � L�� @ * EB6O615 H E:5 � 8 J % 5�8 2B5 H , ;@8�H � @ o % L�5IE�8 , =>5<; J-G � * EB6 � " 8 2:5 .S* ;@8 o %

5IE�8 , =>5<; J-G � " � J 4 0 J EB; =>6:5 , 8/2:5 A * F:F:=>E M � ( H D � L��$� H D � �� � L�� � P H D�� � L�������� t@'42:5 , 5 � H � L(c H � V � " L�"

� 8�=>; J % ?'= J ?B; 8 2 * 8 8/2:=>; A * FBF:=CE M =S; * E * 6:61=C8 =p?N5 =S; J A JN, F:2:=S; A @ ; JO*-.>. 8 2 * 8�4�5 EB5I5<6Q8 J 032:5<0 $R=S;�8/2 * 8�=C8

F , 5<; 5 , ?N57; A ? . 8 =>F . =>0 * 8 = J E %'& F JL. & E J A = *N. ;�" � ML* =CE =C8=S; 5IE J ? M 2 8 J 032:570 $ 8/2 * 8=C8�F , 57;@5 , ?L5<; A ? . 8/=CF . =S0 * 8/= J E

%'& FD* EB6 5 . 5 A 5<E�8/; GK, J A D " � J 8/5 8 2 * 8 � H F � L(c � H Z � L(c�H H Z � L <V H Z � L " L " � J 4 ; =CE 0 5 8/2:5�5IE�8 , =C57; JNG 8 2B5

HB, ;�8OH � @ o % L 0 JL. ? A E ; J-G 8/2:5 .>* ;�8 o % ,/J 4 ; J-G Z *N, 5 *-.>. �I5 , J @ * EB6 8 2:5 .S* ;@8 o % 0 JN. ? A EB; JNG 8 2B5 H , ;@8OH � @ o % L

,/J 4 ; *-, 5 *-. ; J �<5 ,/J @fH Z � L c Z � � +* EB6UH Z � L " c Z� � � ��� � " "� . ; J @ GKJL,�* E & �,Y D @ 0 . 5 *-,/. & � H � � L�c � � H � L�"

��2'?B; � =>; * E D�E FqG =>; J A JN, F:2:=S; A " ��2:5 , 5 GKJN, 5�H D � L�� =>;�=S; J A JL, FB2:=>08 J H D � �� � L�� � P H D�� � L����� � 42:=S032 =>;

=S; J A JL, FB2:=>0�8 JD�E F.G� ' , H F L ) P ��� � P D�E F.G� ' , % H F L ) P]D�E FqG � 'G, % H F L ) 42:=S032D0 J A F . 5 8/5<;�8/2:5+F ,/J�JNG "

T 5+E:5I576 J E:5 A JN, 5�8 2:5 JN, 5 A % 5 GKJN, 5�4�5 0 * ER4 , =C8 5+6 J 4EQ; J A 5�=CE�8 5 , 57;�8/=CE M 0 JN,/JN.>.>*N, =>5<;�" T 5+EB5I5<6O8 J

� W

Page 33: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

; 2 J 4 8 2 * 88/2:5�032 *-,3* 0 8/5 , =>;@8 =S0F JN. & E J A = *-. J-G Z @ GK,/J A 8/2:5 * % J ?L58 2B5 JN, 5 A @�=>;�8/2:5 F , J 61?B0 8 JNG 8 2:5 , ; H F L� ;�"

���b�����"$ �8P"! � 5 8 D % 5 * H 5 . 6 * E 6 . 5I8�?B;�8 * $N5 * E & F JN. & E J A = *N. ; , H F L V�$�$ $�V , % H F L Y D�E F.G

J-G 8 2B5 GKJN, A H @ V L � H F � ahT� tF �

a ��� �MaeT � L " ��2:5IE 615 H E:5 Z� � � ��� 8/2 ,/J ? M 2 Z � � � ����* ;�4 * ;+6 J E:5 =>E

8 2B5 JN, 5 A J!" "���2:5IED8/2:5B� � � A * 8 , = A

Z c ��� Z � � � ��� N $ $�$ NN Z

� � � ��� $ $�$ N$ $�$ $ $�$ $ $�$ $ $�$N N $ $�$ Z

� � � ���

��

2 * ;�032 *-,3* 0 8/5 , =>;@8 =S0F JN. & E J A = *-. J-G , H F L�, " H F L � ��� , % H F L \ 42:5 , 5,\ =>; * EO5 . 5 A 5IE�8 J-G:D " *�-658554X2 � ML* =CE . 5I8

o ; % 5�8 2B5�6:5 MN, 5I5 J-G 8 2:5�F JN. & E J A = *N. , ; H F L "� E:5 A 5I8 2 J 6 8 J�H EB6 615 8/5 , A =CE * E�8/; =>; 8 J�, 5761?B0 5�8 2B5 A * 8 , = A+8 J

5<032B5 .>J E GKJN, A * E 6 8/2:5IE A ? . 8/=CF . & 8 2:55 . 5 A 5<E�8/; *N.CJ E M 8 2:5 61= *-MLJ E *N. 8 JNM 5 8/2:5 , H A ? . 8 =>F . & =>E M %'& * E & ; 0 *-.S*-, ;

& J ? A * & 2 * ?N5�?B; 5<6 8 J , 5761?B0I5�8 2:5 A * 8 , = A * EB6 *-. ; J A ? . 8 =>F . & =>E M %'& @ V GKJL, 54?L5 , & 8/= A 5 & J ? =CE�8/5 , 032 * E M 5<6

8�4 J ,/J 4 ;XL " $1?:F:F J ; 5�4�5 . 5I8 ��c Z @ � �q@B42:5 , 5 � =S;�8 2:5B���A� =S615IE�8/=P8 & A * 8 , = A " T 2:5IEQ0 *-. 0 ? .S* 8/=CE M 8 2B5

615I8 5 , A =CE * E�8 J-G 8 2:=S; A * 8 , =RA =CE 8/2:=>; G * ; 2:= J E 4�50 * E 5 � 570 8 =p?N5 . & 6 J 5 * 032 % .>J 0 $ J-G 8 2:5 A * 8 , = A�H Z� � � ��� @ � � L

; 5IF *N,/* 8 5 . & " ��2:5<E =C8�=S;�0 . 5 *-, 8 2 * 8�8/2:5�615 8/5 , A =>E * EL8 J-G Z�@ � ��=>; � ?B;@8�8 2:5�615 8/5 , A =CE * E�8 J-G Z

� � � ��� @ � �� �

A ? . 8/=CF . =>5<6 %'& 8/2:5 615I8 5 , A =CE * E�8 J-G Z� � � ��� @ � �

� � *E 6D; J J E "���2'?B;�8/2:5 032 *N,/* 0 8/5 , =>;@8 =S0�F JN. & E J A = *-. JNG Z

=S; F . ?B; JL, A =>E�? ; , H F L�, " H F L ��� � , % H F L ; =CEB0I58 2B5 032 *N,/* 0 8 5 , =S;@8 =S0�F JN. & E J A = *-.:JNG Z� � � ��� =>; � ?B;@8�H @ V L � � , ; H F L�"

��2:=S; 0 J A F . 5I8 57;�8 2:5+F ,/J'J-G "

T 5 0 * E G ? , 8 2B5 , 032 *N,/* 0 8/5 , =��<5 8 2B5+F JL. & E J A = *-. ;/, H F L�8 2 ,/J ? M 22, % H F L�"

���b�����"$ �mP�� � G 8 2B5 D�E F.G A J 61? . 5�H D � L � =S; =S; J A JN, F:2:=S0 8 J,D�E F.G� ' , H F L ) P � ��� P D�E FqG � 'G, % H F L ) @

8 2B5IE 8/2:5"032 *N,/* 0 8/5 , =>;@8 =S0DF JL. & E J A = *-. J-G � =S; ; J A 5 5 . 5 A 5<E�8 =CE D A ? . 8 =>F . =>5<6 %�& 8 2B5UF ,/J 61?B0 8 J-G 8 2B5

F JN. & E J A = *N. ; , H F L�8 2 ,/J ? M 2B, % H F L " *�-6585 4X2 � E & D�E F.G A J 61? . 5�H D � L�� @�42:5 , 5 � =S; * E � ��� A * 8 , = A J ?N5 ,

D @1=S;�=S; J A JN, F:2:=S08 JID�E F.G� ' , H F L ) P � ��� P_D�E F.G� 'G, % H F L ) @ GKJN, ; J A 5�F JN. & E J A = *-. ; , H F L8 2 ,/J ? M 26, % H F L %'&

8 2B5 JN, 5 A " " � & J ? , F , 5�?�= J ? ;�8 2:5 JN, 5 A ;�@:=P8 =S; 5 * ; & 8 J ;@5<5�8 2 * 88 2:5 F , J 61?B0 8D, H F L c%�; � , ; H F L A ?B;@8 % 5

?:E:= ��?:5 GKJL,�* E & ; ?B032 G * 0 8 JL, = � * 8 = J ECH ?:F 8 J A ? . 8/=CF . & =CE M %'& * E 5 . 5 A 5<E�8 GK,/J A 8 2:5 H 5 . 6m@'42:=S032 =S; JNG 0 J ? , ; 5

* ?:EB=P8tLt@ ;@=>EB0I5+8 JOM 5 8�8 J J ? , ?:EB= ��?:5 G * 0 8 JN, =�� * 8/= J ED4�5 G * 0 8 JL, 576O8 2:5�, ;XH F L � ; =>EL8 J , 5 .S* 8 =p?N5 . & F , = A 5 F *-, 8/;�"

� J 4�8 2:=S; A J 61? . 5 =>;�=C8/; 5 .CG =>; J A JN, F:2:=S0 8 J H D � L�� 42:5 , 5 Z =S;�8 2B5 % .CJ 0 $U61= *-MLJ E *N. A * 8 , =RA"615<;/0 , = % 576U=>E

J!" J * EB6�J!" � 4=P8/2R; ? %:A * 8 , =>0I5<; Z� � � ��� " ��2:5 , 5 GKJN, 5K@mH D � L � =S;�=>; J A JL, F:2B=>0�8 J H D � L�� @142:=S032 A 5 * EB; Z * EB6

KN

Page 34: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

� *N, 5 ;@= A = .S*-, %'& 8 2:5 JN, 5 A J!" V "��:? , 8 2:5 , A JL, 5M@ * ;�4�5 E J 8 576Q5 *N, . =>5 , @B8/2:5 & 2 * ?L5+8 2B5 ; * A 5 032 *-,3* 0 8/5 , =>;@8 =S0

F JN. & E J A = *N. "���2'?B;�@ ;@=>EB0 5 8 2:5 032 *N,/* 0 8 5 , =S;@8 =S0+F JL. & E J A = *N.�JNG Z =S; 8 2:5 F ,/J 6:?B0 8 J-G 8/2:5�,5; H F L � ; * E 6U; J A 5

H 5 . 6 5 . 5 A 5IE�80@N4�5�2 * ?L58/2 * 8%�; � , ; H F L A ? . 8 =>F . =>5<6 %'& ; J A 5 H 5 . 6 5 . 5 A 5IE�8�=>; 8/2:5 032 *-,3* 0 8 5 , =S;�8/=>0F JL. & E J A = *N.

J-G � "���2:=S; 0 J A F . 5 8 57;�8 2:5+F , J'J-G "

� J 8 5 8/2:=>; *N. ; J F , J ?L5<;�8 2 * 8�8 2:5 615 MN, 5I5 J-G 8/2:5 F ,/J 6:?B0 8 J-G 8 2:5Q, ;XH F L� ;�=>;Q��" T 5 0 * E"E J 4!F ,/J ?N5 *

?N5 , & =>E�8 5 , 57;�8/=CE M 8/2:5 JL, 5 A * % J ?188/2:5 032 *N,/* 0 8 5 , =S;@8 =S0�F JN. & E J A = *-. "

���b�����"$ �8P�� $'?:F:F J ; 5 8/2 * 8�, H F L =S;+8/2:5O032 *-,3* 0 8/5 , =>;@8 =S0 F JN. & E J A = *N.JNG �O@ 42:=S032 =>; ; J A 5

� � � A * 8 , = A J ?N5 ,* H 5 . 6 D " ��2:5<E�,:H �(L�=S;�8 2B5 �I5 ,/J A * 8 , = A " *�-658554X2 � 5 8 D % 5 * H 5 . 6m@ � % 5 * E � � �

A * 8 , =RA J ?N5 ,�*+H 5 . 6 " #�; =CE M 8 2B5 JN, 5 A " 8 2:5 D�E FqG A J 61? . 5lH D � L � =>;�=S; J A JN, F:2:=S08 JID�E FqG � 'G, H F L )-P ��� � P

D�E F.G� 'G, % H F L ) GKJN, ; J A 52, H F L V $�$ $�V , % H F L Y D�E F.G "���2'?B; 8 2B5 , 5O54A'=S;@8/; * E =>; J A JN, F:2:=S; A % 5 8�4�5I5<E�8/2:5 A

42 J ;@5 6 J A * =CE"=>; 8 2B5 D�E F.G A J 61? . 5�H D � L�� . 5I8�? ;+0 *-.>. 8 2:=S;�=>; J A JL, F:2B=>; A � " � & J!" � @ � , ; H F L�c \K, H F L

GKJN, ; J A 5U5 . 5 A 5IE�8�\ Y D " � J 8 =S0 5"8 2 * 8 GKJN,R* E & 5 . 5 A 5<EL8 � Y D�E F.G� ' , H F L )�P � ��� P D�E FqG � 'G, % H F L ) @

, H F L � c N " ��2�? ;�N c � HQN L c � H , H F L � L c ,:H F L � H � L_c , H��(L � H � L " $1=CEB0I5 8/2:=>;D2 JL. 6B; GKJN,R*-.>. ��Y

D�E F.G� 'G, H F L )/P ����� P D�E FqG � 'G, % H F L ) * EB6R;@=>EB0 5 � =S; ; ? , � 570 8/=R?L5K@9,:H �(L ;:c[N "���2'?B;54?L5 , & 0 JL. ? A E JNG ,:H � L

=S; �I5 ,/J @ * EB6 8 2:5 , 5 GKJN, 5 ,:H � L�=>;�8/2:5 �I5 , J A * 8 , =RA "

� 5 8 ?B; 0 J EB0 . ? 615 8 2:=S; ; 5<0 8 = J E %�& 0 J A F *-, =CE M 8/2:5 D�E FqG A J 61? . 5 H D � L � 8 J * H E:=P8/5 � % 5 . = * E MN,/J ?:Ff@

* � A J 6:? . 5�"� J 8/=>0I5+8 2 * 8 8/2:5 * % J ?N5�8/2:5 JL, 5 A 5<;/; 5IE�8 = *-.>. & ;�8 * 8/5<;�8/2 * 8 8/2:5 032 *N,/* 0 8/5 , =>;@8 =S0�F JL. & E J A = *-. J-G

* A * 8 , = A =S; 8/2:5 ��� J�� � J-G 8/2:5!H D � L � A J 61? . 5 =CEU8 2:5 ; * A 5 4 * & * ;�8 2:5 JN, 615 , J-G�* H E:=C8 5 � % 5 . = * E ML, J ?:F

�� ��� �

P ��� � P �� � � �

=S;-, � � , " ��� � ��� , % � � "�� J 8 =S0 5�8 2 * 8 GKJN, 54?L5 , & H E:=C8 5 � % 5 . = * E MN,/J ?:F 4�5�2 * ?L5 * ; ; J 0 = * 8/5<6

4=C8 2D=P80@ * ?:EB= ��?:5�?:E JL, 6:5 , 5<6 ; 5���?:5<EB0 5 J-G F , = A 5 =>E�8 5 M 5 , ; ,/* =>; 5<6 8 J F J 4�5 , ; 8/2 * 8 0 J A F . 5 8 5 . & 6:5 8 5 , A =>E:5<;

8 2B5�;@8 , ?B0 8 ? , 5 JNG 8 2B5 ML, J ?:F "��:? , 8 2B5 , A JL, 5M@L5�?N5 , & 8�4 J H EB=P8/5�� % 5 . = * E MN,/J ?:F ; 8/2 * 82 * ?L5 8 2B=>;�; 5���?:5IE 0 5 J-G

=>EL8/5 M 5 , ; *-, 5�=>; J A JN, F:2:=S0�" � JL, 8 2B5 D�E FqG A J 61? . 5�H D � L��O4�5 2 * ?N5�8 2:5 ; * A 5 ;@=C8 ? * 8 = J E 5�A:0I5IF18 8/2 * 8�4�52 * ?L5

* ?:E:= �L?B5 ?BE JN, 615 , 5<6";@5 ��?:5IEB0I5 = ,/, 5761?B0 = % . 5 F JN. & E J A = *-. ; ,/* =>; 5<6R8 J F J 4�5 , ; 8/2 * 8 0 J A F . 5 8 5 . & 6:5 8 5 , A =>E:5<;

8 2B5 ;@8 , ?B0 8/? , 5 JNG 8 2:5 A J 61? . 5�"�$1=CEB0I58�4 J�D�E FqG A J 61? . 5<;�@sH D � L ��* EB6�H D � L �

�*-, 5=>; J A JN, F:2:=S0�= G * EB6 J E . &

= G � �* EB6 � " * ; = A = .S*-, @ =C8 =S;�5 * ; & 8 J ; 5I5 8 2 * 8 GKJN, 54?L5 , &OA * 8 , = AD8 2:5 , 5 =>; * ?BE:= ��?:5 ;@5 ��?:5IEB0I5 JNG F J 4�5 , ;

J-G = ,/, 5761?B0 = % . 5 F JN. & E J A = *-. ; * ;/; J 0I= * 8 576R4=P8/2U=C8�; ?B032Q8 2 * 8�8�4 J A * 8 , =>0I5<; *-, 5 ;@= A = .S*-, = G�* EB6 J E . & = G 8/2:5 &

V

Page 35: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

2 * ?L5 8/2:5 ; * A 5 ; 5���?:5<EB0 5 J-G F JL. & E J A = *N. ;�" � =>E *-.>. & E J 8 =S0 5 8 2 * 8 * A * 8 , = AU=S;+61= *-MLJ E *N. = � * % . 5 = G*-.>. 8 2B5<; 5

F JN. & E J A = *N. ;�2 * ?N5�615 ML, 5<5 V "

� E ; ? A A *N, & @ 4�5 2 * ?N5 F ,/J ?N576Q8/2 * 8 42B5IE B =S; * F , =CE 0 =>F *-. =S615 *N. 6 J A * =CEf@ 54?L5 , & ; ? %:A J 61? . 5 J-G*

H E:=C8 5 . & M 5IE:5 ,/* 8 5<6 B A J 6:? . 5=>; H E:=C8 5 . & M 5<E:5 ,3* 8/5<6 " T 5�2 * ?N5�F , J ?L5<6�8 2 * 8�42B5IE B =>; * �?B0 . =>6:5 * E 6 J A * =>E

4�50 * E G * 0 8 JN, * E & H EB=P8/5 . & M 5IE:5 ,/* 8 576 A J 61? . 5?BE:= ��?:5 . & * ; * A J 61? . 5 J-G 8/2:5 GKJN, A B � 'G, � � ) P ����� P B � 'G, % � �

42:5 , 5�8 2B5 , ; � ; *-, 5�= ,/, 5761?B0I= % . 5 " T 5+2 * ?N5 *-. ; J F ,/J ?N576 8/2 * 842:5<E�,:H F L�=>;�8/2:5 032 *-,3* 0 8 5 , =S;�8/=>0�F JL. & E J A = *N.

J-G�* A * 8 , =RA �O@'8/2 * 8 ,:H �(L�=S;�8 2B5 �I5 ,/J A * 8 , = A "

��

Page 36: A CHARACTERIZATION OF FINITELY GENERATED MODULES by

�I= � �Q=Q5��s- +�� � �

� *-.>. = * E8@ �NJ ;@5<F:2 � "-� J E�8 5 A F JL,/*N, & � % ;�8 ,/* 0 8 � .>M 5 % ,3* " �N8 2 576 "���5I4 � JL, $ ( � J ? M 2�8 J E Q=�� =CE � J A F * E & @

V7WNW � "

� * & @�� * ?'=>6 ��"�� =CE:5 *-, � .CM 5 % ,3* * EB6D=P83;� F .>. =>0 * 8 = J E ; " �-EB6 576 " �5 * 61=>E M * ;/; * 032'?B; 5 8@83; ( ��6:6:=>; J E!

T 5<; . 5 & @ V7WNW � "