HEREDITARY RADICALS AND QUASLRADICALS IN UNIVERSAL …poeschel/poePUBLICATIONSpdf/1984_PoeHereditar… · HEREDITARY RADICALS AND QUASLRADICALS IN UNIVERSAL ALGEBRAS R. PÖSCHEL (Berlin)

Acta Math. Hung. 44 (1-2) (1984), 35-59.

HEREDITARY RADICALS AND QUASLRADICALS IN UNIVERSAL ALGEBRAS

R. PÖSCHEL (Berlin)

Dedicated to Dr. habil. Hans-Jürgen Hoehnke

Introduction

The theory of radicals, first developed for rings, has since been extended to various other algebraic structures. There is also a general radical theory for arbitrary universal algebras, which goes back to H. J. Hoehnke [7]. A radical in the sense of Hoehnke assigns to each algebra 2i of a given class (e.g., a variety) a congruence relation R(2I) so that

(Rl) f ( R ( ' % ) ) s ~ ( f (2I)) for any homomorphism f from 2I, and

Specializing this notion to rings, however, one arrives at radicals which are more general than the classical KuroS-Amitsur radicals. A direct generalization of the latter to universal algebras has been given by R. Mlitz [15], [16].

In the present paper we consider radicals in the sense of Hoehnke which are hereditary with respect to subalgebras :

(R3) R(W)= R(2I) f l W X W for subalgebras W of %.

In case of rings these radicals prove to be radicals in the sense of KuroS- Amitsur, too. Note that hereditary radicals are not the Same as the usually considered radicals with hereditary radical class (for hereditariness properties of radical or semisimple classes See, e.g., Stewart [19], Wiegandt [20], Leavitt [l 11).

Heredity is a very strong property (in case of rings it is called strong heredity), which enables to give a full description of all hereditary radicals in arbitrary varieties of algebras. Keeping the strong condition of heredity we can, on the other hand, wea- ken the notion of radical: applying the Same methods we get a description of all hereditary quasiradicals (i.e. "radicals" without (R2)) in universal algebras.

The approach presented here is - in some sense - a generalization of Maranda's

-& description of hereditary (quasi-)radicals in the variety Mod S of unitary right modu- 1, les over a ring S [13]. Maranda assigns to every (quasi-)radical R the Set M of all

(right-)ideals B of S such that S/B is radical. Conversely, assigning to a set M of : right ideals (= S-submodules) of S the mapping R defined by R(2I)= {aE2iJker(ls-

»U)€ M). Maranda gets all hereditary (quasi-)radicals starting with Sets M of speci- fied properties (here (Is-a) denotes the uniquely determined S-module-homomorphism S-2I which sends the unit lsES to U€%, %E Mod s).

In general; there is no 1-1-correspondence between subalgebras and congruence relations (kernels of homomorphisms). This leads to the distinction between (quasi-) radicals and 0-(quasi-)radicals (the latter assign to each algebra 2I a subalgebra r(2I)). Because of heredity, the algebras generated by one or two elements determine

3. Acta Mathematica Hungarica 44 , 1984

the 0-(quasi-)radicals or (quasi-)radicals, resp., at all. Such one- or two-generated algebras can be represented as homomorphic images (or, equivalently, as congruences) of F, or F„ the free algebras (in a given variety) with one or two generators, resp. In fact, this was done by Maranda since S can be considered as the free S-module with one generator (cf. 8.4). Now in the general setting, instead of S, we shall use F, or F„ and the above sets M of right-ideals (in case of Mod S ) correspond to subsets (so called Q-sets or R-sets) of Con F, or Con F, (the congruence lattices of F, and F,).

The present paper consists of 7 sections. In $1 we give some preliminary definitions and notations. In $2 we introduce an important technical tool for our investi- gations (namely Q-sets and R-sets). $3 consists of the main result of this paper: Every hereditary radical or quasiradical in a variety V can be uniquely characterized in terms of a subset of Con F,. In particular, (quasi-)radicals can be presented by quantifier-free formulas of the first order predicate calculus provided that F, is finite. In $4 semisimple and radical classes of hereditary radicals are investigated and characterized. As an example, hereditary (quasi-)radicals in the variety of S-systems are considered in 95, and all radicals are listed for a concrete semigroup S.

Throughout the last three sections we suppose that our variety admits a constant fundamental operation whose value 0 is a subalgebra. $6 deals with hereditary O- quasiradicals and characterizes them in terms of subsets of Con F,. Hereditary 0-quasiradicals are uniquely determined by their "radical" classes, which are also characterized. In $7, hereditary 0-radicals are considered. If there is a "satisfactory" connection between congruences and 0-classes ("normal subalgebras"), then hereditary 0-radicals can be characterized analogously to 0-quasiradicals. Finally, $8 specializes the results of $6, 97 to modules and rings. For unitary left (or right) modules over a ring S , specialization of our results yields essentially those of Maranda [13]. We also give an explicit classification of hereditary (quasi-)radicals for abelian groups (i.e. Z-modules) by means of ideals in the lattice (N; g.c.d., 1.c.m.). In the variety of rings, hereditary radicals are A-radicals in the sense of Gardner [3], i.e., they are already determined by a radical defined on the additive groups of the rings. Therefore the characterization of hereditary radicals in the variety of abelian groups provides at the same time a characterization of hereditary ring radicals, too. For associative rings, however, one gets another characterization.

The investigation of hereditary radicals in universal algebras was suggested to the author by H. J. Hoehnke more than 10 years ago. Most of the results of the present paper have already been obtained in 1970171. Encouragement by L. Mhrki and R. Wiegandt during the author's stay in Budapest in 1981 finally led to the publication of these results. The author thanks J. H. Hoehnke, L. Mhrki, R. Mlitz and R. Wie- gandt for stimulating discussions and many useful hints. The author also thanks the referee for many useful cntical remarks, which led to an improved version of the manuscript.

$1. Preliminaries

1.1. Let V be a variety (equational class) of universal algebras. V is fixed throughout the next sections. For a given algebra %=(A; (h)i,I)EY let Q,

B

be the set of all n-ary fundamental operations of 2i and let Q= U Q,. We write n=O

X€% if X is an element of the base set A of %. The property of %' being a subalgebra

Acta Mathematica Hungarica 44, 1984

HEREDITARY RADICALS AND QUASIRADICALS 37

of (U will be denoted by (U'S%. Con (U denotes the congruence lattice of %€Y. The trivial congruences are 0=0,= {(U, a ) J a ~ A) and 1 = I,= AXA.

1.2. We recall some well-known notations: For a map f: A-B we have

ker f = {(U, b) J f (U) = f (b)) (kernel of f ),

f (A)=Imf={f(a) la€A) (imageof f) ,

For a congruence 0ECon (U (%€Y), [U], denotes the equivalence class of 0 containing aEA (the index 0 will not be mentioned if it is clear from the context), 1,: (U- -(U/0: a-[U], denotes the natural homomorphism onto the factor algebra (U/0.

1.3. Fora Set X, let Fv(X) - for short F(X) - be the Y-free algebra in Y generated by X, in particular let F,= F(e) and F,= F(e„ e,) be the Y-free algebras generated by the elements e and e,, e,, respectively. Let 1],1,„ be the unique homomorphism from F, into an algebra (U€ Y which makes the following diagramm commutative :

F(e1, e,) \ "01, aa Y ' % \

The kernel ker qYL,„ will be denoted by BN(al, U,) or sometimes by the more impres- sive notation ker, (e,-U,, e,-U,) (the index '2i will be deleted if (U is clear from the context). Defining a(a„a,):=~]li,,,(a), the elements aEF(el,e,) - notation a=a(e„ e,) - can be interpreted in every (U€ Y (in particular in F, itself) as binary term functions a: A2-A : (U,, U,)-a(al, U,). Then O(al, U,) can be presented as

ea(a1, U,) = ker% (e, - al , e2 - a2) =

1.4. Note that Im tf1, „ is exactly the subalgebra (U,, U,), of (U generated by a, and U,. Thus

(U,, 4% = Cl, a2 (F2) 2 He1, ez)/@'(al, U,).

52. Q-sets and R-sets

Hereditary (quasi-)radicals will be characterized by so-called Q-sets and R-sets, which are subsets of Con F,. In this section we introduce these notions by defining operations on Con F, under which Q-sets and R-sets have to be closed,

2.1. Let 0, be the congruence of F(e„ e,) generated by identifying e, and e,:

e i d = kerFC4(el - e, e2 - e) = {(a(ei, ej), el))(aEF2; i, j, k , IE{1, 2)).

We have F(e) 2 F(el, e,)/Bid (cf. 1.4).

Acta Malhematica Hungarica 44, 1984

2.2. For BECon Fz, let B- be defined by "interchanging" the roles of e1 and e,:

2.3. Let B1,B2ECon F,. We embed B, and 0, into F,= F(e„ eh e,) as follows (B2 is considered as a congruence of F(e„ e,), e, and e3 playing the roles of el and e,) :

6; = ~C;,eZ(ei) = {(~(e l , ez), ß(e1, e , ) )€~; / (u(e~ , e2), ß(el, eZ))€ Bi),

e2* = 11;;,e3(e2) = {(U(%, 4 , ß(e2, e3))€~,2!(u(e1, e2), ß(el, eZ))€ß,).

Let B* be the congruence of F, generated by B: U B:. We define B, o e2€Con F, by

2.4. Let fE Cl,, n s 1, and let B,, .. ., e,€Con F,. Now consider Bi as an element of Con F(eli, eZi) and embed B,, ..., B, into 2I:= F(el1, eZl, ..., eli, e 2 ~ , ..., eln, e2,) (all eji are to be different). One gets

n

Let B* be the congruence of 2I generated by U Bi. We define f [B1, ..., Bn]ECon F2 i=l

by f [B, , . . . , B,] = ker<~/e* (el H [alle*, e2 H [a21s*) = g'le* ([alle*, [azls*),

where U , = f (ell, . . . , eh) and a, = f (e„ , . . . , e„). If f is unary (n= 1), then f [B] can be given more explicitly as follows:

2.5. For L S C o n F, and BECon F, we define

For short, let (0: (a, 8)) := ker~~le(ei [a1oI e2 [ßle)

(in generalization of a ring-theoretic notation). Thus

(Lle) = {(U, ß) €F: ((B: (U, ß ) ) ~ L).

We collect some properties of the above defined notions in the following proposition, the straightforward proof of which easily follows from the definitions and will be omitted.

A c t a Mathe ina t ica H u n g a r i c a 44, 1984

HEREDITARY RADICALS AND QUPSIRADICALS 39

2.6. PROPOSITION. Lei %€^Y, 8,B'ECon F„ a,ai,ajE% (iE{l, ..., n}), fEQ, (nz i ) , u=u(el,e2)EF, and ß=P(el,e,)EF,. Then

d) (B: (a, P)) = (6: (P, U>)-,

e) (0: (U, P)) = eFz(u, P), ( 1 P ) = 1, (U, P > E ~ eid G (0: (U, P)),

2.7. DEFINITION. A subset L G Con F(e„ e,) is called a Q-sei, if L satisfies the following conditions :

(LO): (LOa) BidEL,

(Ll): BEL, B G B' * B'EL (for B'ECon F,);

L is called an R-sei, if it, in addition, satisfies the following condition:

(L3) (L(O)EL*BEL (for BECon F,).

2.8. REMARKS. a) Conditions (L0)-(L3) are chosen in such a way that the relation {(U, b)E9i2ker(e1-a, e,-b)EL} (which, for each %, we shall assign to I, in $3) becomes an equivalence by (LO) and (Ll), a congruence by (LI) and (L2) and a radical by (L3).

b) Note that for condition (L3) the set (Lid) of pairs has to be a congruence relation. We shall see later (cf. 3.4b) that (LIB) is always a congruence for Q-sets L.

C) While (Ll) works "upwards" in the lattice Con F,, condition (L3) goes "down- wards", since BS(LI9) for Q-sets L (due to 2.6e and (LOa), (Ll)). Moreover, if 0 ' s B and if L 5 Con F, satisfies (Ll), then (Lid') G (LIB) by 2.6h and 2.5. There- fore, in order to check property (L3) for L, it suffices to check (Lid) $L only for all BE T of a "covering" subset T of Con F2\L satisfying VB'ECon F2\L3BE T: B' G B. In fact, (LI9')EL would imply (Ll9)EL by (Ll).


93. Characterization of hereditary quasiradicals and radicals

3.1. DEFINITIONS. Let Y be a variety and let R be a mapping which assigns to each % € Y a congruence R(2i)ECon 2i. R is called a quasiradical (e.g. [7, p. 3751) in Y if

(Rl) g(R(2i))g R(g(2i)) for every hornornorphism g frorn %€Y.

R is called a radical (in the sense of Hoehnke, [7, p. 3561) if it satisfies in addition

(R2) R (%IR(%)) = O for all % € Y

A (quasi-)radial R is said to be hereditary if we have

(R3) 2i '~2i*R(W)=R(2i)nWXW

for all 2i, WEY. Let Qrady and Rady be the sets of all quasiradicals and radicals in Y, respectively.

REMARK. A hereditary quasiradical R satisfies

W)' g (R(%))SR(B)

for all hornornorphisrns g : 2i-B(%, BEY). Now we are going to define two operations R and L which will establish a con-

nection between the subsets of Con F, and the mappings R : A- R (A).

3.2. DEFINITIONS and REMARKS. Let L 5 Con F„ and R: 2i-R (2i) be a rnapping defined on Y such that R(2i) 5 %X 2i is a binary relation. Then we define a rnapping RL: 2i»RL(2i) &%X% and a subset LR 5 Con F, as follows:

= {(U, b) c2i2)OpI(a, b) E L) for 2i €"Y; (2) .

LR := {ker 3,ECon F21(3,(e3, 3,(e2))€ R(~,(F,)), 3, hornornorphisrn from F,) =

Suppose now R(2i)ECon 'U (e.g. in case REQrady) then LR can also be defined as follows: (2)' LR = {OECon F2(Oid 5 HR(0)),

where HR(0) is the cornplete preirnage of R(F,/O) in F„ i.e.,

a) We have L = LRL for all L 5 Con F, (what directly follows frorn the definitions).

b) Condition R=RLR is equivalent to heredity of the mapping R, i.e. to the property

V% EY : R (2i) = {(U, b) E a2)(a, b) ER(@, b)d).

Acta Mathematica Hungarica 44 , 1984


In fact, R=RLR implies RLR(B)= {(a, b)lB5(a, b)€LR)= RLR(%)nB X B for B I%, since B5(a, b)=Ba(a, b) for a, b c B (cf. 1.4, (a, b),= (a, b ) ~ ) . Conversely, for hereditary R we have (a, b)E R(%)o(a, b)€ R(%) C7 ((a, b),#= R((a, b )%)o o(A(e1),R(e2))€R(R(F2)) for I=& (cf. 1.4)okerAELRo(a,b)CRLR(2l).

It turns out that Q-sets (R-sets) and hereditary quasiradicals (radicals) correspond to each other under R and L. We have:

3.3. CHARACTERIZAT~ON THEOREM. Let L Con F,. Then the foiiowing conditions are equiz)alent:

(i) L is a Q-set (R-set, resp.), (ii) there exists a hereditary quasiradical (radical, resp.) R in Y such that L = LR, (iii) RL is a hereditary quasiradical (radical, resp.) in Y, (iv) RL is a quasiradical (radical, resp.) in Y. Let R : ITI- R (B) be a mapping dejined on Y such that R (%) & % X %. Then

the following conditions are equivalent: (i)' R is a hereditary quasiradical (radical, resp.),

(ii)' there exists a Q-set (R-set, resp.) L 5 Con F, such that R= RL, (iii)' R = RLR and LR is a Q-set (R-set, resp.).

PROOF. We are going to prove (i) *(iv) *(iii) =+(ii)*(i) in case of quasiradicals : (i)=+(iv). Let L be a Q-set (see 2.7). Then RL is a quasiradical. In fact, RL(%)=

= {(a, b)jBa(a, b ) ~ L) is a congruence relation because RL(%) is reflexive, symmetric and transitive by (LO), (Ll), (2.6a) and (2.6b), and RL(%) is a congruence by (L2), (LI) and (2.6~). Moreover, RL fulfils 3.1 (Rl) since Oa(a, b) 5 05(g(a), g(b)) for a, bE% and any surjective homomorphism g : %-B, i.e. (by (LI)), ~ (RL(%)) 5 RL(8).

(iv)=+(iii) follows from RL= RLRL (3.2a) and 3.2b. (iii)-(ii) trivially follows from L = LRL. (ii)*(i). Let L= LR for a hereditary quasiradical R. Then R= RL because of

3.2b. We show L is a Q-set: Condition (LOa) trivially holds (See e.g. 3.2(2)'). Condi- tion (Lob) is fulfilled because ([eile, [ezle)E R (Fz/B) implies ([e2]e-, [eile-)E R (F,/B-) (use 2.2 and (Rl)). To show condition (LOc) we use the notations given in 2.3: let Bi, B2ELR, i.e., ([e1lei, [e2le,)€ R(F2/Bi) (i= 1, 2). This implies (by 3.1(Rl)', cf. 2.3): ([eile*, [ezle*>E R (F3/8*) arid ([ezle*, [e3le*)E R (F3/8*). Thus ([el], [%])ER (F3/B*) = =RL(F3/B*), i.e., B,~B~=0~31~*([e,] , [e3])EL and we are done in showing (LOc).

Now, L fulfils (Ll). In fact, let BEL, B&B'. Then, by (Rl), ([eile, [e2]e)ER(F2/B) implies cp ([e,]„ [e21e) = ([eile,, [e21et)E R (F2/B') for the canonical mapping cp : F2/B - -F2/B'. Consequently B'€ LR = L.

L fulfils (L2). Let B,, ..., B,€ L, fE S Z , (n 2 1) and ai= Fz/Bi, thus ([e1lei, [e2],,)E R(%,), 1 i i i n . Using the notations given in 2.4, this implies ([elile., [e2i]e*)E R(%/B*) by 3.1 (Rl)'. R(%/B*) is a congruence, therefore ([alle*, [az]e*)E ER(%/O*) for al= f(ell, ..., e„), a2= f(e2,, .. . , e„). Thus (note 2.4, 3.2(1) and R = RL), f [B,, . . . , B,] = Oale* ([a,], [a2])E L and we are done.

It remains to prove the above equivalences in case of radicals. Because every radical (R-set, resp.) is also a quasiradical (Q-set, resp.) we have to show for hereditary quasiradicals R only that L = LR satisfies (L3) if and only if R= RL satisfies

Acta Mathematica Hungarica 44 , I984

(R3). This will be done in the next lemma (3.4~). Therefore the first part of 3.3 is proved. Concerning the second part,

(i)'*(iii)' follows from (ii)*(i) and 3.2b, (iii)'*(ii)' is trivial, (ii)'=(i)' follows from (i)-(iii).

Therefore all conditions (i)'-(iii)' are equivalent, too. D

3.4. LEMMA. Let L SCon F, be a Q-set und R= RL the corresponding hereditary quasiradical (cf. 3.3). Then we Iiave:

a) FIR(LU)([alR(a), [bIR(%)) = (LlOLU(a, b)) for a, b~ b) (LO)= HR(B) for OECon F, (cf. 3.2), C) L is an R-set if und only R is a hereditary radical.

b) follows from a) and (2.6g) because O'IR('U)([a], [b])= HR(6) in case PI= F2/B, a = [eile, b = [e,l,.

C) Assume L=LR is an R-set. Then R= RL is a hereditary quasiradical by 3.3 and satisfies (R3). In fact,

eWiR(%) ([U], [b]) = (L 1 0% (U, b)) E L (U, b) E L * a)

Conversely, if R is a hereditary radical, then the Q-set L = LR also satisfies (1,3). In fact, let (LO)EL for BECon F 2 . Then B=Oa(a, b) for appropriate a, bE%CY (cf. 2.6g) and (LIBN(a, b)) O"lR('U)([a], [b])E L implies ([U], [ b ] ) ~ R('U/R(%)) = 0 (since R = RL). Consequently, (U, b)E R(%)= RL(%), i.e., O=BB(a, b)EL. n

3.5. For a given (Q- or) R-set L, the corresponding (quasi-)radical R= RL can be expressed more explicitly. Let L'= { ~ , ( i ~ I ) be a subset of L with the property V8E L3iEI: Oi& 8. Then R(%)= {(U, b)cQ12/ 3iEI: Bis 0"(a, b)) (cf. 3.2(1), 2.7(Ll)). But this gives (cf. 1.3)

R ( a ) = {(U, b ) ~ % ~ 1 3 i E f ~ ( a ( e ~ , e,), ß(el, ez))EOi: a(a, b) = ß(a, b)).

This description can be very useful in many special cases; in particular, R(%) is presented by a quantifier-free first order formula provided that F, is finite (see also 5.2).

Acta Mnthematica Hllngarica 4 4 . 1984


54. Semisimple and radical classes

For technical reasons we have to distinguish the class Y0 of all 1-element algebras in the variety Y. Every radical R in a variety Y is determined by its semisimple class (cf. e.g. [7, Korollar 251)

via R((U) = (RY)((U), where

We have R = RY'R for RERady. On the other hand, the radical is not, in general, determined by its radical class

(The algebras in Y0 are considered to be radical algebras.) Sometimes, however, B(R) uniquely determines R, e.g. in case of Kuroi-Amitsur radicals for groups or rings.

It is natural to ask how one can characterize semisimple and radical classes of hereditary radicals in V . In this section we shall treat this question. The answer will be given in terms of R-sets (it is an Open problem to give a more satisfactory answer in terrns of the elements of the semisirnple or radical class only). We begin with a necessary condition for Y'R :

4.1. PROPOSITION. Let R be a kereditary radical in Y. Then Y = 9 R satisfies the follo wing conditions:

(i) Y is closed under subdirect products, (ii) Y is hereditary (i.e. closed under taking subalgebras),

(iii) (U€ Y if (and, by (ii), only ifl (a, b) 4 (RY)((a, b),) for all a , bE (U, a # b. Equivalently, (U€ Y if and only if for all a # bc (U there exists a homomorphism cp from (a, b)41 into an element of Y such that (a, b)gker cp.

REMARKS. Property (i) is known to characterize completely the semisimple classes of (arbitrary) radicals in Y , in particular, (i) implies Y = 9 R Y (See e.g. [7, Kor. 251). Condition (ii) provides "half-heredity", namely it implies R((U')s R((U) for (U' 5 (U, which in turn gives RLR((U) 5 R((U) (as it can be easily seen by checking the definitions). We also note that (iii) implies the following condition (provided that (i) and therefore Y = 9 R Y holds) :

(iv) (UEY ifand only ifevery 2-generated subalgebra of (U (i.e. generated by two elements) belongs to Y.

The straightforward proof of 4.1 is omitted.

4.2. For a given class Y 5 Y we define

L Y := {BE Con F,l (e„ e,)E B' for all B'E Con F, with 8 5 B' and FdB'E Y).

In view of R Y (F,/B) = n {B'/B I F,/B'E Y ) we have :

L Y = (8 Con F, 1 ([eilc, [e,le) E RY(F2/B)) = LR9' = {B E Gon F 2 1 e i d H R ~ (B)}

(cf. 3.2(2), (2)').


4.3. THEOREM. Let Y V. The following conditions are equivalent: (a) Y is the semisimple class of a hereditary radical in V , (b) Y satisfies 4.l(i)-(iii) und L Y is an R-set (cf. 4.2, 2.7), (C) Y satisfies 4.l(i)-(iii) und L Y satisfies (LO) und (L2), (d) Y satisfies 4.1 (i)-(iii) und RLY 2 RY.

Moreover, under these assumptions we haue R Y = RLY, Y = YRLY und L Y = =LRY.

PROOF. (a)*(b). The first part of (b) follows from 4.1. Now, L9'=(q.2)LRY is an R-set because R Y is hereditary and 3.3(i)'=(iii)'.

(b)*(a). We shall show Y=YRLY, then we are done because RLY is a hereditary radical by 3.3(ii)'*(i)'. We have

= { ( U , b) E 212 1 Oa(a, b) E LY) =

This, however, implies

(a)o(d). If Y satisfies (a), R Y is hereditary, thus RLY =RLRY = R Y by 4.2 and 3.3(iii)'. Conversely, if Y satisfies (d), then R Y 5 RLY = RLRY R Y (see remark to 4.1). By 3.2b), RLRY = R Y is hereditary, i.e., Y =YRY satisfies (a).

In order to prove 4.3 it remains to show (c)=+(b) since (b)-(C) is trivial. Thus assume (C). One directly derives from definition 4.2 that (Ll) holds for LY, therefore L Y is a Q-set. Then (LY 119) = HRLy (B) by 3.4b. As above one gets RLY = RLRY 5 5 RY (by 4.2 and the remark to 4.1). Due to 4.l(i), Y is the semisimple class of the radical RY, therefore HRy is a closure Operator as shown e.g. in [7; Satz 38, p. 3721. Now, let (LY IB)= H R L ~ (B) be an element of LY. Then, by definition 4.2,

which gives BELY, i.e., L Y satisfies (L3), too; therefore it is an R-set and (b) is proved. U

Now what can be said about the radical classes of hereditary radicals? How to characterize them and how to recognize whether different hereditary radicals have the same radical class? We are going to investigate these questions and begin with proving some necessary conditions.

4.4. PROPOSITION. Let R be a hereditary radical in V . Then the radical class X = B(R) satisfies the following conditions:

(i) X is homomorphically closed und V0 X , (ii) X i s hereditary (cf. 4.l(ii)), (iii) 2 l c X if und only i f every 2-generated subalgebra of <U belongs to X.

The proof is straightforward and will be omitted. @

REMARK. 4.4(i) characterizes radical classes of arbitrary radicals [9; p. 205(9.1)].

Acta Mathematica Hungarica 44,1984

HEREDlTARY RADlCALS AND QUASlRADlCALS 45

4.5. For a radical R in Y , let (R) be the least hereditary radical containing R (i.e., R(%)c (R)(%) for all %€Y) . For L& Con F„ let (L) be the least R-set containing L. This R-set (L) exists because the intersection of R-sets can easily be shown to be also an R-set;

4.6. PROPOSITION. Let XSY be a class of algebras satisfying 4.4(i)-(iii) und L,:= {BECon F, I F,/BE X ) . Moreover, let R = R(Y\X) be the radical with semisimple class Y \X . Then (R) = R(LR)= R(L,) und (R) is the least hereditary radical the radical class of which contains X .

PROOF. Due to 4.4(i) (and the definitions), we have R(%)= 0 for % ( X and R(%)= 1 for %EX, i.e. B ( R ) = Z Therefore R is the least radical with B(R) 2X and (R) is the hereditary radical claimed in the proposition.

At first we show RLR 2 R. In fact, R(%)# 0 gives R(%)= 1 and, by 4.4(ii), R ( a f ) = l for all 2-generated 'Ws%. Thus BW(a, b)ELR for all a, bE%, i.e., (RLR)(%)= 1. Therefore R(LR) 2 RLR 2 R is a hereditary radical containing R. consequently (R) 5 R(LR). Conversely,

(R) = fl {R'ERady I R' hereditary, R R'} =

=„.„ fl {RL'J L' is an R-set, R RL') =

=(3.2(1))R( fl {L'IL' is an R-set, R 5 RL')) 2 2 - R ( fl {L'IL' is an R-set, LR LRL' = L')) = R(LR),

i.e., (R) = R(LR). It remains to prove (LR)= (L,). Indeed,

4.7. THEOREM. Let Xc Y und L,= {BECon F21F~BEX). Then the following conditions are equivalent:

(a) X is the radical class for (at least one) hereditary radical in Y, (b) X satisfes 4.4(i)-(iii) und ((L,)IB)= 1 implies BE L, (for BECon F,), (C) X=B(R(L,)) (cf. 4.5, 3.2(1)).

PROOF. (c)=+(a) is trivial and (a)=+(c) follows from 4.6 (in view of 4.4). (c)=+(b). The first part of (b) follows from 4.4. Moreover, we have

((L,) 18) = 1 a H R ( L ~ (B) = 1 (cf. 3.4b) a

o R(L,) (F2/B) = 1 a &/BE X a B E L,, (C)

(b)=t(c). Assuming (b), for % € Y we have to show R(L,)(%)= 1=+%€ X (the opposite implication directly follows from 4.6). R(L,)(%)=l implies R(L,)(W)= 1 for all2-generated subalgebras %' of %. Such an %'=(U, b), is isomorphic to F2/B where B = BW(a, b) (cf. 1.4), therefore (as above) HR(Ld (B)= = ((L,)JB)= 1. By (b), we get BE L,, i.e., W z F,/BEX for every 2-generated subalgebra W. Due to 4.4(iii), we have %EX.

4.8. PROPOSITION. Let R be a hereditary radical in Y und L,= {BECon F.1 I(LR(B)= 1). Then R'= R(L,) is the least hereditary radical which has the same


radical class as R. Two hereditary radicals R, und R, in Y haue the same radical class if and only if

(LRllO) = 1 o (LR210) = 1 for all OE Con F,.

PROOF. By 4.6 and 4.7 we are done with the following observation: F,/OEX'= = W(R)oHR(0)= l a ( L R O ) = 1 (oOELo), (cf. 3.4b, note R= RLR). n

Finally we can ask the question how to describe (R), the least hereditary radical containing R (a special case was treated in 4.6). The next proposition provides an answer if R already fulfils some properties of a hereditary radical.

4.9. PROPOSITION. Let R be a radical in Y, the semisimple class Y = 9 R of which fulfils 4.l(iii). Then

(R) = R(LR) = R(LY) (cf. 4.2).

PROOF. Let R'= R(LR). We have

Rf(%) 2 RLR(9I) = {(a, b) (Oa(a, b) E LR} = {(a, b) 1 (U, b)E R((a, b)~)}.

Thus, R'(S[)=O implies R(S[)=O via 4.l(iii), i.e., 9 R ' G 9 R , which gives R 5 R'. Let R" be a hereditary radical containing R. Then R"= RL"? R for some R-set L" and we get L"=LR7'2LR. Thus L"rj(LR) and RM=RL"ZR(LR)=R'. Conse- quently, Rf=(R).

4.10. PROBLEM. It seems that the radical class W(R) of a hereditary radical R does not uniquely determine the radical R. The author, howc-\er, does not know an exarnple of two hereditary radicala with the same radical classes.

Problem: 1s there a variety Y and R such that Rf R' (for notations See 4.8)? For 0-quasiradicals See 6.6.

$5. Hereditary (quasi-)radicals of S-systems

This section shall serve as an exarnple to demonstrate the results of the preceding sections in case of a very special variety - the ~ariety of S-systems.

Let S be a fixed semigroup with unit 1. A (unitary) S-system (cf. [6], [8]) is an algebra (M; (s)„~) such that for each SES there is a unary fundamental operation

satisfying ml =m and (rris)s.'=tn(ss') for all mEM, s, s'ES (ss' denotes the multiplication in S ) .

Let Y be the variety of all unitary S-systerns. We are going to describe all hereditary (quasi-)radicals in Y u~ ing Theorem 3.3. At first we must therefore investigate the structure of F(el, e,) and its congruences.

5.1. The *Y-freely generated S-system on 2 generators is sirnply the disjoint union of two copies of S:

Acta Mathematicc Hungarica 44 , 1984

HEREDlTARY RADICALS AND QUASIRADlCALS 47

where the fundamental operations are naturally defined by (eis)sf=ei(ssf) (i= l , 2 ; s, sfES).

Note that for m,, m2E NEY, the congruence Oa(ml, m,) (see 1.3) looks as follows :

0'u(mlm2) = kera(el W m,, e2 W m,) = {(eis, ejsf)€F,21mis = mjsf (i,jE{l, 2))).

5.2. REMARK. If S is finite, F, and Con F, are finite, too. Therefore the number of hereditary quasiradicals is also finite. Moreover, every hereditary (quasi-)radical R in Y can be described by a finite quantifier-free first order formula (cf. 3.5). In fact, let R= RL for a Q-set L (cf. 3.3) and let L'= {B,, ..., B,) be the Set of all mini- mal congruences in L. Then

R(%) = {(ml, m2)E%21Bl 5 Oe(ml, m,) or. ..or B, S Oe(ml, m,))

for % € Y , or, more explicitly (see 3.5):

5.3. Now it is not difficult to describe explicitly the operations Bi„ B-, BoBf, s[B], (B: (a, ß)) and (L (B) for B, BfECon F, (e.g., s[B]= {(eisf, ejsf7J(eissf, ej.~sf')EB)), See 2.1-2.5, but we shall not go into technical details. Then, according to 2.7, it is quite clear what Q-sets and R-sets are, and these in turn characterize hereditqry quasiradicals and radicals for S-systems. A concrete example will be given below.

REMARK. J. K. Luedeman [12] investigates torsion theories for S-systems via so-called left quotient filters in S . There seems to be some connections (not clarified yet) to our characterization of hereditary radicals via R-sets, in particular in case of 0-(quasi-)radicals (cf. $6, $7) of S-systems.

5.4. As an example, from now on let S= ((1, a, b); .) be the following semigroup (1 is the "unit", b is a "zero"):

F(e)= {el, ea, eb)=S has 3 congruences (we describe them by the corresponding partitions of F(e)) :

eo(e) = {{el), {Ca), W ) ) = 0,

B; (e) = {{el, ea, eb)) = 1.

5.5. There are 21 congruence relations in F(el, e,), namely 9, 9 ,2 and 1 of type (I), (2), (3) and (4), respectively:


( 1 ) Q,=e:(e,)Ue;(e,>, X , Y € (0, a, 1);

where 8:(ei) is equal to the partition 8:(e) (see 5.4) when e is substituted by ei;

(2) 8:, = (8, U {e, b, e, b ) ) = the congruence generated by 8, and identifying e, b and e2 b ( X , Y € (0, a, 1 )) ;

(3) 8?d={{ell, ezl), {ela, e2a), {e,b, e,b))=Oid (cf. @ I ) ) , G= {{eil, ela, 4, e2a), {e, b, P&));

(4) B'= {{eil , eza), {ela, e21 ), {elb, e2b)).

It is an easy but tedious task to see that these relations are congruences and that there are no others. The congruence lattice Con F, is given in Fig. I .

Fig. I

5.6. Let us compute explicitly the operations on Con F, given in 5.3:

(i) eid = 8yd,

(ii) 8, = O Y x , (8iy)- = 8:- (8fd)- = 8Td, (8')- = 0°,

(iii) 8, o 8,1yr = 8„,, 8, V 8iPy, = 8,, , 8, o 8id = 8x,(y

8, o 8' = 8„ 8iY o 8:.,, = 8iy. , 8 f y o 8fd = 8;.

B i y V 8' = 8iY , 8fd o 19;; = 8f:,+", 8yd o 0' = 0°,

8Td V 8' = 8yd, 8' V 8' = 8yd,

Acta Mathematica Hunga~ica 44, 1984

HLREDlTARY RADlCALS AND QUASIRADICALS 49

where X, (0, a, l), zc (0, a) and x+y=max (X, y) with respect to the ordering O-=a-= I . The operations 8' V 8 not listed here can be obtained frorn (ii) via 8' V 8 = =(8- o(8')-)-.

(iv) s[0] = 8 for OECon F, and sE (1, U),

for 0 = 8„ b[O] =

otherwise.

5.7. With 5.6 we are ready to find all Q-rets (see 2.7). By the symmetry of 5.6(ii) cornbined with 2.7(LOb) and by 5.6(iii) combined with 2.7(LOc), the Q-sets L rnust be "generated" (i.e., using (Ll) and ( L o ~ ) ) by 8' and/or by elements 8 lying on the verti- cal middle axis of Fig. 1, in notation : L = (8)Q. Thus there exist exactly the.following 10 Q-sets :

= { B I B 31 ei,l) = {eyd, es, e:,), L; = L, U {oO),

L ~ = ~ ~ ~ Q ~ ~ ~ ~ ) u L ~ = ( ~ „ ) ~ , L;=L,U{BO),

L3 = (8 10 2 8Aa) U Li = (8A0)Q, La = LS U {B0),

L 4 = { 8 ( @ 2 8 „ ) U L 1 = ( 8 a a ) ~ , L ~ = J L U { ~ ~ ) ,

L5 = {O 10 2 Go) = (oUo)~,

L, = (810 2 OO0) = (O,,,,), = Con F,.

5.8. The hereditary quasiradicals Ri= RL, corresponding to the Q-sets described in 5.7 can be characterized asfollou~s (we use the characterization described in 5.2), BI€ Y :

&((U) = {(ml, m,) 1 m, = m,) = 0 (trivial radical),

R3((U) = {(ml, m,) I m, = m2V (ml = mla A m, = m2aAm, b = m, b)),

R,(BI) = {(m,, m2)lml = m2V(ml = n~,aAm, = m2a)),

R,(PL) = 2I X (U = 1 (trivial radical),

R; ((U) = {(m„ m,) 1 m, = m2Vm, = m,a),

4 Acta Mothemalica ffungarica 44, 1984

5.9. It turns out that all quasiradicals in 5.8, except Ra and Ri, are radicals, too. Thi5 can be shown by proving that the corresponding Q-sets are (or are not) R-sets. But we can also directly show the radical property 3.1(R2). In fact, e.g. for R, we have

3 (m, = m, a Am, = m2 a)V[m,] = [m21 =+ (ml, m2)E R4(%)

Analogously one gets R(~I/R(%))=O for RE {Ri(l 5 i s 6)U {R;, RB). However, Rj and Ri are not radicals as shown by the following example. Let 'U be the S-system generated by m, and m, such that m,b=m,b (See Fig. 2). Then (m,, m,)(R(%) but ([m,], [m2])E R (%IR(%)) for RE {Ra, RI), i.e., R (%IR(%)) # 0.

Fig. 2

Fig. 3 shows the lattice (with respect to inclusion) of all hereditary quasiradicals in the variety of all S-systems where S is the semigroup given in 5.4 (the radicals are marked with full dots).

6

R l

Fig. 3

66. 0-quasiradicals and their characterization

The ideas and methods used for the characterization of (quasi-)radicals also apply to the characterization of 0-quasiradicals (the case of 0-radicals is treated in the next section). Definitions, results and proofs for 0-quasiradicals are similar to those of quasiradicals. Therefore, in this section, we develop the corresponding notions and results very briefly.


HEREDlTARY RADlCALS AND QUASlRADlCALS 51

6.1. From now on let % be a variety of universal algebras such that there is a constant fundamental operation whose value, say 0, is a subalgebra. To avoid some technical modifications we shall assume that there are no other constants except 0 (i.e., IS1,)= 1). Let Sub % be the lattice of all subalgebras of %€"L;;. Note that OE%' for each %'€Sub %.

6..2. DEFINITIONS. A map r: %»r(%) which assigns to each %E% a subalgebra r(%)ESub % is called 0-quasiradical (cf. [10]) in "L;; if

(RO1) g(r(%)) r (g(%)) for every homomorphism g from %€"L;;.

A 0-quasiradical r is called hereditary if for all 2l', %E%:

(R03) %'s2I*r(%')=r(%) n %';

in particular we have r (r(%))= r(%).

REMARK. A hereditary 0-quasiradical also satisfies

(RO1)' g(r(2I)) 5 r (b ) for every homomorphism g : %-.B.

We shall characterize hereditary 0-quasiradicals by means of subsets of Con F(e) - so-called Qo-sets which we are going to define now.

6.3. DEFINITIONS and NOTATIONS. In F,= F(e) (the "L;;-free algebra on one generator), we define analogously to 1.3

za (U) : = ker = {(CL (e), ß (e) ) E F? J CL (U) = ß (U)).

where is the unique homomorphism which makes the following diagramm commutative :

F(e)

Fl/rQ(a) is isomorphic to (U)% - the subalgebra of % generated by a. Let fE G?,, ( n ~ 1) and Tl, ..., 7,ECon F,. We embed zi into %= F(el, ..., e,)

(considering f i as an element of Con F(ei)) as follows:

T' = V: (7i) = {(U (eil, ß(ei)) E %" (CL(e), ß ( e ) ) ~ 7i).

n

Let T* be the congruence of % generated by U 7;. We define f [TI, . . ., z,]ECon F, by i = l

f [ T ~ , . . . , T,,] = ker (F, -;u' % - %/T*) = zal+ ([U],*) 'I, Ar*

where U = f (e„ . . ., e,). Now, a subset 1 s Con F, is called a P-set if it satisfies

(LOO) lEl,

(LOl) 7 EI, 7 5 7' * 7'E 1 (for T'€ Con F,),

(L02) 71, ..., z,EI*f[zl ,..., z,,]El (for fEQ„ n z 1).

49 Acta Mathematica Hungarica 44 , 1984

REMARK. If IQol - 1 then (LOO) must be modified into zw(c)E I for every cE Qo . 6.4. DEFINITIONS and REMARKS. Let 1 C Con F, and let r : 'U-r('U) be a

map defined on -L;; with r('U) & 2l. We define a subset k Con F, and a rnap rl: 'U-rl('U) as follows:

(1) rl('U) := { a ~ ( U ( z ~ ( a ) ~ l ) ,

(2) Ir := {T E Con F(e) I [el, € r (~ (e l l~ )} .

If r('U)ESub 'U for 'U€% then we have

a) l r l=l holds always (follows frorn the definitions). b) A function r with r('U)ESub 'U is hereditary (equivalently, r fulfils r(%) =

= {a~'Ulr((a),)=(a)~)) if und only if rlr=r. In fact, r=r l r irnplies r(B)=rlr(B)= = {aEB(rB(a)~l r )= rlr( 'U)nB=r('U)nB for 235% since .r"(a)=rW(a) for aEB (cf. 6:3). Conversely, for hereditary r we get rlr(c21)= {a€'U(rm(a)E lr)= {aE 'U1 I~(F,/T"(~))= F1/Tm(a))= {aE 91 I r((a),d= (a),)=r((21).

Now we are ready to formulate the characterization theorern for hereditary O- quasiradicals.

6.5. THEOREM. Let r und 1 be as in 6.4. (a) r is a hereditary 0-yuasiradical in 6 ifand only if Ir is a Qo-set und r = rlr. (b) 1 is a Qo-set if und only if rl is a hereditary 0-yuasiradical.

PROOF. (b) Let 1 be a Qo-set, then rl fulfils 6.2(R01) due to 6.3(L01). Moreover, rl('U) is a subalgebra of 'U. In fact, OE rl(2l) since zw(0)= 1 E 1, and for U,, . . . , an€ E rl(c21), fE Q,,, we have za(al), . . . , za(a,)E 1, consequently f [za(al), . . . , zm(a,)] Cza(f(al, ..., an))€/ (due to (LO1) and (L02)), i.e., f(al, ..., a,)Erl('U). Finally, rl is hereditary because of rl= rlrl and 6.4b). Conversely, if rl is a hereditary 0-quasiradical, then I=lr l is a Qo-set by (a).

(a) Let r be a hereditary 0-quasiradical. Then r = rlr by 6.4b). We show that Ir is a Qo-set. Condition (LOO) is satisfied since l=zF(e) (0)~l r due to OEr(F(e)). Condition (LO1) easily follows from 6.2(R01). To show (L02), let T„ . . ., T,€ lr and f E Q, (n& 1). With the notations given in 6.3 one gets ai:=[ei],*Er(BI/r*) by (RO1)', thus [U],*= f (U,, . . . , an)€ r('U/r*)= rlr('U/~*). Therefore f [T,, . . . , T,] = zal"([a],*)€ lr. The converse irnplication of (a) directly follows frorn the above proof of (b).

For a 0-quasiradical r in -L;;, the set

is called the radical class of r, the elernents of which are said to be radical algebras. The following theorern characterizes such radical classes. Moreover, it turns out, that hereditary 0-quasiradicals are cornpletely deterrnined by their radical classes.

6.6. THEOREM. (A) A class X26 is the radical class of a hereditary O-quasiradical if und only if X satisyes the following conditions:

Acta Mathemnfica Hungarica 44, 1984

HEREDlTARY RADICALS AND QUASIRADICALS

(i) X 'is homomorphically closed, (ii) X is hereditary, (iii) the union of radical subalgebras is again radical. (B) Zf X is the radical class of a hereditary 0-quasiradical r, then r is uniquely

defined by 4%) = {aE'LI((a)aEX).

(DifJerent hereditary 0-quasiradicals have dzrerent radical classes.) Moreover, r ('LI) is the greatest radical subalgebra of 'LI E%.

REMARK. For X a s in (B), I= {t€Con Fl(Fl/zEX) is a Qo-set.

PROOF. (B) follows directly from heredity (cf. also 6.4b)), in particular, for 'LI'€ X , 'LI' 5 'LI, we have W = r(2I') = r ('LI) n 'LI' G r (2l).

To prove (A), let X = B ( r ) for a hereditary 0-quasiradical r. Then conditions (i)-(iii) are easily checked ((iii) follows from (B)). Conversely, if X satisfies (i)- (iii), then r ('LI): = {a€'LII( U),€ X ) defines a hereditary 0-quasiradical ; in fact, r(%) is a subalgebra of 2l due to (ii) and (iii); 6.2(R01) follows from (i) and (R03) is obvious (or See 6.4b)). U

7.1. In order to define a radical property "R ('LI/R(X))=O" for 0-quasiradicals in % one needs a connection between subalgebras and congruences. For BECon 'LI and B E S U ~ 'LI, 'LIEK, we define

0°= {xE'LIl(x, 0)EO) (=[0]„ the 0-class of O), - B = f' {OE Con %(B 0°) (the congruence "generated" by B).

We have a very good situation e.g. for abelian groups or rnodules (F= 0 and gO= 23 for 935%) or for rings (@=O and S0=!B for ideals 23 of 'LI) but, clearly, these properties do not hold in general.

7.2. DEFINITIONS. A 0-quasiradical r in % (cf. 6.2) is called 0-radical in % if it satisfies

(R02) r('LI/f('LI))= (0) for all 'LILIE%,

where f (BI): = r(2i) is the congruence "generated" by r(2i). We remark that a O- quasiradical r is a 0-radical if and only if F is a radical in %.

A 0-(quasi-)radical is called 0*-(quasi-)radical if it satisfies

We shall See that there is a satisfactory characterization of hereditary 0-radicals only if the congruence F(%) defines the 0-radical r(%) uniquely by means of (*), i.e., if r(X) is a "normal subalgebra" of 'LI.

REMARK. (*) trivially holds e.g. in the variety of abelian groups and also in any variety of S-systems (cf. 55) with Zero.

Acta iV1athematica Hungarica 44.19g.f

Having 6.5 in rnind, now it is natural to ask which conditions a Qb-set 1 has to satisfy in order to define not only a O-quasiradical rl (via 6.4(1)) but also a O-radical or a O*-radical. The following definitions and results shall prepare the answer.

7.3. DEFINITIONS. A Qo-set 1s Con F, is called Q*-set, if rl satisfies 7.2( *) (i.e., is a O*-quasiradical). A Qo-set (or Q*-set) is called Ro-set (or R*-set) if the following condition holds :

(L03) ( I l ~ ) € l = > TEI for rECon F,, where

(1 1 T) := {(uie), ßie))€ ~ ( e ) ' ] ( ~ ( 4 , ß(a)) E r n ) ) = ([al)

with cU= F(e)/r, a=[e],. The definition of a Q*-set I is not very satisfactory because it does not use proper-

ties of I only. In the most important varieties, however, the property 7.2( *) of being the O-class of a congruence can be expressed by closure properties with respect to fixed terrn functions. E.g., in varieties of groups or rings we have: A subgroup (or subring) B z cU is a normal subgroup (or an ideal, resp.) if and only if f (b, x)E B and f(x, b)EB for all X € % , bEB where f(x, y)=x-lyx in case of groups and f(x, y)=xy in case of rings. The following proposition Covers all these cases and can be used for a "satisfactory" characterization of Q*-sets.

7.4. PROPOSITION. Let 6 be a variety with 0, fE Q, and iE {I, . . ., n). A hereditary O-quasiradical r in ^Y;; satis$es

if and only if the corresponding Qo-set 1 = Ir satisfies

COROLLARY. I n case of rings, Q*-sets are characterized by 6.3(L00)-(L02) and

(L02)* VTE I: f [T, OIE 1 and f [0, T]€ 1 where fc Q, is the ring multiplication.

PROOF. Let (2)* be fulfilled. Then a,E rl(%) = r (cU) irnply t"(aj)€l for , j€ (1, ..., i ) . But . .

for arbitrary a,, ... , an€ 'U (cf. 2.6c), thus tw(f(al, . .., an))€/ due to (LO1) and (2)*, consequently f (a, , . . . , an)€ rI(cU)=r(%), i.e., (I)* is satisfied. Conversely, assume r= rl satisfies ( I )* for a Qo-set I. We use the notations given in 6.3: if ~ ~ € 1 (jc (1, .. . . . . , i)), then [e],,€ rr(F,/~,), thus [eil,*€ rl(Bt/~*) by 6.2(R01)'. Hence

[ ~ I P = [f (e19 . . . , en)l,* = f ([eil, . . . , [enl) E r l (W*) by (I)*. Thus

f [T,, ..., T ~ , q+, , ..., T,,] = tB~r*([a]r*)~I,

in particular for Ti+,= ... =T,,=O.

7.5. LEMMA. Let I be a Q*-set and sECon F,. For u(e)EF, define

(7: (4 ) = zF1"([u (41,) = {(ß1(e), ßz(e)) ((ß1(u (e)), ßz(a(e)))€ T).

Acta Motheaalica Hungwicd 44,:11984


Then = {a(e)E~,I(r: a(e))El).

REMARKS. We have (T : a (e))" = (e) lß (U (e)) E TO} and (zm(a) : a (e)) = =zm(a(a)) for every aE2lE"L;;. In the variety of rings, (T: a(e))O is the so-called Tschirnhaus-transformation of the ideal TO with a (cf. [18; $122, p. 4681).

PROOF. Let 2l and a be as in 7.3. Then

7.6. For a Qo-set I, the 0-quasiradical rl satisfies rl(W)& W2nrl(21) for W s 2l (due to the heredity of rl). The following condition claims equality for single- generated W :

(* *) r l ( ( ~ ) ~ ) = (U); fl rl(2l) for all a E 2l E "L;; . This condition is equivalent to

(* *)' z m r ~ ([U]) = (I 1 zm (U)) for all a E 2l E "L;;,

since the left and right side of (* *)' are equal to

and

respectively (cf. 7.3, 6.3). The next lemma will show that this condition holds under various other (stron-

ger) conditions. For the theorem below, however, we shall need (* *)' only.

7.7. LEMMA. Condition 7.6(* *) is satisjied if e.g. (i) I is a Q*-set und F= T for all .rECon Fl, in particular for varieties "L;; of

abelian groups, modules or rings, or if (ii) "L;; is a variety of S-systems with 0 (cf. $5).

PROOF. (i) rl(2l) n W 2 is a congruence on 2l' for WS%. Therefore

But this implies ( * *) since P=T. (ii) Condition (* *) can be checked directly using the definitions.

REMARK. Case 7.7(ii) can be generalized to all varieties "L;; in which the property (al , U,)€ 2l' for WS %€"L;; can be expressed by a quantifier-free first order formula in which besides term equations only predicates f (al, U,)€%' can occur where f is some term function.

Now we are ready to characterize hereditary 0-radicals.

7.8. THEOREM. (a) Zf r is a hereditary 0*-radical in "L;;, then Ir is an R*-set und r = rlr.

(b) Zf I& Con F, is an Ro-set (R*-set, resp.) satisfying 7.6(* *), then rl is a hereditary 0-radical (O*-radical, resp.) und I= lrl.


REMARK. A comparision with 6.5 shows that the characterization of 0-quasira-. dicals went off more smoothly than those of 0-radicals. However, Theorem 7.8 gives a 1-1-correspondence between 0*-radicals and R*-sets, if, e.g., rO=z for all rECon F, (because every R*-set then satisfies ( * *), cf. 7.7). Moreover, 7.8 provides a 1-1-correspondence between 0-radicals and Ro-sets if, e.g., in addition %=!B for b 5% holds in % (because ( *) is then always fulfilled).

PROOF OF 7.8. (a) Let r be a hereditary 0*-radical. Then r is in particular a O- quasiradical, thus r= rlr and Ir is a Qo-set by 6.5(a). Moreover, Ir is a Q+-set due to Definition 7.3. We show 7.3(L03). Let zECon F,, %= F,/T, a=[e],. Then

(b) Due to 6.5(b) it remains to show 7.2(R02) for r l where I is a Qo-set satisfying 7.6(* *). In fact, let U€%€%. Then

Thus rl(%/rl(%))= {[O]). ii

98. Hereditary (quasi-)radicals of abelian groups and rings

8.1. Let Ab be the variety of all abelian groups. In view of the l-l-correspondence between subgroups and congruences there is no distinction between (quasi-) radicals, 0-(quasi-)radicals und 0*-(quasi-) radicals.

We use the following notations: For F(e) in Ab we take Z (generated by e= 1). A subgroup U i Z is generated by a uniquely determined element uEN= = (0, 1,2, . . .); the corresponding congruence U is denoted by x(u) (x(u)O= [O],(„= =(U)). Thus ( N ; ( ) z (Con F(e); 2 ) (where u (V:-u is a divisor of V) and X 1s the isomorphism. Therefore we can describe subsets of Con F(e) by means of subsets of N. One can imagine that this simplifiesconsiderably the description of Q-sets and hereditary (quasi-)radicals. Reformulating the notions defined in $6 we get the following :

zG(a) = x(ord(a)) for aEGEAb, where

0 if na # 0 for all n W 0, ord (U) =

min {nl na = 0, n W O), otherwise;

for U, vEN and the fundamental operations o - E 0,: X+- -X, o + E Q2: (X, Y)+-xfy (see 6.3). Moreover, we have

HEREDITARY RADICALS AND QUAS~RADICALS 57

(since ( % ( U ) : z)O= {~(ujwz) , cf. 7.5) for uEN, zEZ; in particular x(p)El implies x (pm)c ( l ~ ( ~ " + l ) ) for a prime number p and a QO-set (=Q*-set) l&Con F(e), m~ l (since (x(~"+ ' ) : p " )=x(p )~ l , thus p " ~ (llx(pm+l))O, cf. 7.5). These observations lead to the following theorem:

8.2. THEOREM. Fora subset [ S N , rl(G)= {a~Glord ( a ) ~ / ) defines a hereditary (0-)quasirrrclical or radical, respectively, in Ab if a l ~ d only if I satisfie.~

or, respec,tively, in addition

(L03) p E I=+pmE / for prinie nulnbers p und for ~ I E N.

Convcrsely, i f r is a Iiereditary (0-)quasiradical (radical, resp.) then

(where Zn= Z/x(rz)) fulfils (LOO)-(L02) ( (Lo~) , resp.) ancl r = rlr.

PROOF. The proof follows irnmediately from 8.1, 6.4, 6.5 and 7.8. The only thing we have to be sure is the equivalence of 8.2(L03) (for / S N ) and 7.3(L03) (where I has to be substituted by %(I)= {%(U) / u ~ l ) ) ) .

In fact, the implications pE l=+x(p)E x(l)=+x(pm)E X(/) (by (LO1), 8.1 and induction on ni)=rpm€l show the above conditions to be necessary. Conversely, let 8.2(L03) be fulfilled, let x(w) := (x(l)(x(u))€ x(1) for some uEN and let x(u) 4 X([). We have to find a contradiction. By 8.2(L03) and (L02) there also exists a prime number plu with p (I. Then p{w (otherwise x(w) & x(p) Ex(/) by (~'1)) and pl(u/g.c.d. (U, W)). Therefore (ulg.c.d. (U, W)) 41 (by (LOI)), i.e., (%(U): w) 4 x(l) (cf. 8.1). Hence (cf. 7.5) W (f (x(l)lx(u))O= x(w)O, a contradiction.

8.3. In view of (L03), hereditary radicals r in Ab are uniquely defined by a subset P' of P= {plp prime or P= 1) or by P'= {O)U P as follows: r=rp,:= rl ccf. 8.2), where

I = [(P') := { npq i 1 I finite, piE P', ai€IV). i E I

Tl~us mery hereditary rad i~a l is the direct surli of its prinie components:

P GEAb, PEP'

where rp(G) = (acG13nEN: ord ( U ) = pn).

(REMARIZ. This is the converse of a result of S. Dickson [I] that the sum of "hom-orthogonal" families of torsion subfunctors is again torsion.)

For P'= P we get the classical torsion radical in Ab:

rp(G) = {a~G13n€N\{O): na = 0).

The trivial radicals can be obtained from P'= {I) and P'=PU (0).

A c t a Matheiilnticci FIungai'ica 44, 1984

8.4. REMARKS. a) The hereditary (0-)(quasi-)radicals in the variety Mod S of unitary left-modules over a ring S (with unit) can be treated analogously to the variety Ab. In this case, Theorem 3.3 (or,equivalently, 6.5 and 7.8) lead to the results of J. M. Maranda [13 ; Proposition 4, p. 11 91,. where (L02) for " +" and the module operations is equivalent to Maranda's conditions (G2) and (G3) and

(L is a Qo-set of left ideals of S , t is a left ideal of S ) is equivalent to Maranda's condition (G4) [13; Proposition 3, p. 1191.

If S is a commutative principal ideal ring with no proper Zero divisors then we get for Mod S the same characterization as in case of Ab=Mod Z.

b) In every variety of not only commutative semigroups with unit (monoids) or groups there do not exist non-trivial hereditary 0-quasiradicals (one can prove, e.g., W[T, T] = 0 for z€ Con F(e), z # 1, where W denotes the operation of multiplication; therefore only trivial Qo-sets exist). The results (for monoids) also follow from the fact that every monoid can be embedded into a congruence-free monoid as observed in [14; Theorem 31 (since then F(e) is either radical or semisimple, consequently I -generated subalgebras are all either radical or semisimple, i.e., the hereditary radical is trivial).

8.5. Let now Y be the variety Ring of all rings (not necessarily associative) or the variety aRing of all associative rings. The notion of a radical for universal algebras in the sense of Hoehnke (cf. 3.1) is - when specialized to rings - much more general than the classical notion of radical in the sense of KuroS-Amitsur (see e.g. [2]). Since congruences in rings are uniquely determined by their 0-classes, i.e., by ideals, there is no distinction between radicals und 0*-radicals (7.2) in Y. Moreover, because of 6.6(B), the radical classes of hereditary 0*-radicals are also radical classes in the sense of KuroS-Amitsur. (But we have to choose 0*-radicals and not O-radicals in order that r(2l) be an ideal and not only a subring.)

Consequently, hereditary 0*-radicals in Y are precisely the Kuroi-Amitsur radicals which are hereditary with respect to subrings (in ring theory these radicals are called strongly hereditary).

Thus, Theorem 7.8 (und 7.7, 7.4) will give a complete description of strongly hereditary KuroS-Amitsur radicals via R*-sets (i.e., subsets of F(e) satisfying 6.3 (LOO)-(L02), 7.4(L02)* and 7.3(L03)). To get a more explicit characterization of such R*-sets it remains to reformulate the conditions to be satisfied in terms of ideals (or their parameters) of ~ ( e ) = { z aieilui# 0 for finitely many i). This, however,

i = l

would exceed the aim of this Paper. But for Y = Ring, we shall See that the determination of R*-sets can be avoided

because the whole problem can be traced back to the variety Ab.

8.6. Let r be a hereditary radical (or, equivalently, a 0*-radical) in Ring. Then the semisimple class of r is hereditary (4.l(ii)). Therefore we can apply a result of B. J. Gardner [5; Corollary 2.51 (cf. [4; Theorem 2.41) which gives that the radical class B ( r ) of r is a so-called A-radical[3], i.e., W(r)= {2l€Ringl%+€B(r+)), where 21f is the additive group of the ring (U and r + is the radical in Ab defined as the restriction of r to allzero rings. Thus hereditary radicals in Ring are in 1-1 -correspondence with hereditary radicals in Ab: r(%)=r+(%+). From the results of Gardner it follows

A c t a M a t h e m a t i c a H u n g a r i c a 44, 1964


that r+(21t) is an ideal in every ring b with B+=%+. This property can also be directly Seen from 8.2. In fact, ord (ax) and ord (xa) are divisors of ord (U) for any elements a, X of a ring %= (A; f , .), thus rl(%+)= {aESllord (a)El) is an ideal by (LO1).

With Theorem 8.2 we already know all hereditary radicals in Ab. Due to the above observations, the following proposition is obvious:

8.7. PROPOSITION. Theorem 8.2 remains valid in case of radicals i f A b is substituted by Ring.

It is not clear whether the Same holds for quasiradicals, too. But in this case, of Course, one can use Theorem 6.5 and 7.4, in order to get a characterization of O*- quasiradicals (or, equivalently, quasiradicals) in Ring or aRing.

With 8.7 and 8.2 we have a complete description of all strongly hereditary radicals in R i n g in the sense of Kurog-Amitsur.

References

[I] S. Dickson, Direct decomposition of radicals, Proc. Conf. on Categorical Algebra. La Jolla 1965. Springer Verlag (Berlin, Heidelberg, New York, 1966), pp. 366-374.

[2] N. J. Divinsky, Rings und radicals (Toronto, 1965). [3] B. J. Gardner, Radicals for abelian groups and associative rings, Acta Math. Acad. Sci. Hungar..

24 (1973), 25S268. 141 B. J. Gardner. Some cumnt issues in radical theorv. Math. Chronicle. 8 (1979). 1-23. [5j B. J. ~ardner,'Some degeneracy and pathology in non-associative radical theoj; Annales Univ.

Sci. Budapest Eötvös Sect. Math., 22/23 (1979/80), 65-74. [6] H. J. Hoehnke, Uber das untere und obere Radikal einer Halbgruppe, Math. Z.. 89 (1965),

300-311. [7] H. J. Hoehnke, Radikale in allgemeinen Algebren, Math. Nachr.. 32 (1966), 347-383. [8] H. J. Hoehnke, Einige neue Resultate über abstrakte Halbgruppen, Colloq. Math., 14 (1966),

329-348. [9] H. J. Hoehnke, Die Radikale und das Prinzip des maximalen homomorphen Bildes in Bika-

tegorien, Arch. Math. Brno. 3 (1967), 191-207. [10] H. J. Hoehnke, Das Brown-McCoysche 0-Radikal für Algebren und seine Anwendung in

der Theorie der Halbgnippen, Fund. Mathematicae, 66 (1970), 155-175. [ l l ] W. G. Leavitt, Strongly hereditary radicals, Proc. Anier. Math. Soc., 21 (1969), 703-705. [12] J. K. Luedeman, Torsion theories und semigroups of quotients ( I , 11). Clemson University, techni-

cal reports No. 297 (Jan. 1979) and No. 321 (July 1979). [13] J.-M. Maranda, Injective structures, Trans. Amer. Math. Soc., 110 (1964), 98-135. [14] L. Mhrki, R. M:litz, Free monoids and strict radicals, Studia Sci. Math. Hungar.. 13 (1978),

225-228. [15] R. Mlitz, Kurosch-Amitsur-Radikale in der universalen Algebra, Publ. Math. Debrecen, 24

(1977), 333-341. [16] R. M:litz, Moduly i radikaly universal'nyh algebr, Zzv. Vuzov, ser. Matem., 3 (1977), 77-85

(Russian). [17] R. Mlitz, Radicals and semisimple classes of Q-groups, Proc. Edinburgh Math. Soc., 23 (ser.

11) (1980). 3 7 4 1 . [18] L. Rkdei, ~&ebra"(l. Teil) (Leipzig, 1959). [19] P. N. Stewart, Strongly hereditary radical classes, J. London Math. Soc. (2), 4 (1972), 499-509. 1201 R. Wiegandt, Semisimple classes and H-relations, Studia Sci. Math. Hungar., 13 (1978), 181-

185. (Received April 7 , 1982; revised May 23.1983)

INSTITUT FÜR MATHEMATIK DDR-1080 BERLlN MOHRENSTR. 39

Acta Mathematica Hungarico 44, 1984

83-4304 - Szcgedi Nyomda - F. V.: Dob6 Jdzsef igazgat6

Documents

HEREDITARY RADICALS AND QUASLRADICALS IN UNIVERSAL …poeschel/poePUBLICATIONSpdf/1984_PoeHereditar… · HEREDITARY RADICALS AND QUASLRADICALS IN UNIVERSAL ALGEBRAS R. PÖSCHEL (Berlin)