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Invited lecture on CHARACTERIZATION OF NEAR-FIELD SPACES
OVER BAER-IDEALS
Dr. N V Nagendram1
Professor
Department of Mathematics Kakinada Institute of Technology & Science Tirupathi(V), Divili, Peddapuram(Mandal),
East Godavari District 533 433 Andhra Pradesh. INDIA.
E-mail: [email protected]
Abstract
An ideal I of a near-field space N is called a right Baer-ideal if there exists an idempotent e N such that r(I) = eN. We know that N is quasi-Baer if every ideal of near-field space N is a right Baer-ideal, N is n-generalized right quasi-Baer if for each I N the ideal In is a right Baer-ideal, and N is right principally quasi-Baer if every principal right ideal of a near-field space N is a right Baer-ideal. Therefore the concept of Baer ideal is important and it got extended to near-field spaces.
In this paper, I investigate some properties of Baer-ideals and give a characterization of Baer-ideals in n n generalized triangular matrix near-rings, full and upper triangular matrix near-rings, semi prime near-ring and near-ring of continuous functions in a defined near-field space N. Finally, I had equivalent conditions for which the n n generalized triangular matrix near-ring be right SA.
Subject Classification Code: MSC (2010):16D25, 54G05, 54C40. Keywords: Near-ring, Near-field, Near – field space, Quasi-Baer ring, quasi near-field space over Baer-ideal, Generalized right quasi-Baer, Semi-central idempotent, Spec(S), Extremally disconnected Near-field space. _____________________________________________________________________________ Corresponding author: Dr N V Nagendram E-mail Id: [email protected]
Section 1: Introduction
Throughout this paper, N denotes as Near-field space has zero symmetric near-ring with identity. Let X N. Then X N denotes that X is an ideal of N. For any subset T of N, l(T) and r(T) denote the left annihilator and the right annihilator of T in near-field space N. The near-field space of n-by-n (upper triangular) matrices over near-field space N is denoted by Mn(N) (Tn(N)). An idempotent e of a near-field space N is called left (right) semi-central if ae = eae (ea = eae) for all a N. It can be easily checked that an idempotent e of near-field space N is left (right) semi-central if and only if eN (Ne) is an ideal. Also note that an idempotent e is left semi-central if and only if 1e is right semi-central. See [17] and [9], for a more detailed account of semi-central idempotents.
Thus for a left (or right) ideal J of a near-field space N, if l(J) = Ne (r(J) = eN) with an idempotent e, then e is right (left) semi-central, since Ne (eN) is an ideal, and I use Tl(N) (Tr(N)) to denote the set of left (right) semi-central idempotents of near-field space N.
In depth study makes me to characterize near-field spaces over baer-ideals in near-field spaces by defining quasi-Baer near-field space [2]. Near-field space N to be a quasi near-field space over Baer-ideals if the left annihilator of every ideal of N is generated, as a left ideal, by an idempotent. I used the quasi-Baer concept to characterize Near-field space N over Baer-ideal when a finite-dimensional algebra with identity over an algebraically closed near-field space is isomorphic to a twisted matrix unit semi-group algebra. The quasi-Baer condition are left-right symmetric. It is to find that Near-field space N is a quasi-Baer if and only if Mn(N) is quasi-Baer if and only if Kn(N) is a quasi near-field space over Baer-ideal. ([18],[10],[11] and [3] for Quasi Baer- ring).
A near-field space N to be n-generalized right quasi- near-field space over a Baer-ideal if for each J N, the right annihilator of Jn is generated (as a right ideal) by an idempotent. A near-field space N is n-generalized quasi near-field space over a Baer-ideal if and only if Mn(N) is n-generalized. Moreover, Dr. N V Nagendram found equivalent conditions for which the 2 2 generalized triangular matrix near-field space be n-generalized quasi near-field space over Baer-ideal ([4], Theorem 4.7 and 4.3)
A principally quasi near-field space over a Baer-ideal is introduced by
Dr. N V Nagendram and used them to characterize and generalize many results
on reduced (i.e., it has no nonzero nilpotent elements) p.p.-near-field spaces. A
near-field space N is called right principally quasi-Baer (or simply right p. q.-
Baer) if the right annihilator of a principal right ideal is generated by an
idempotent.
The characterization of near-field spaces over Baer-ideal is studied
and results obtained by Dr. N V Nagendram. An ideal J of N is called right
Baer-ideal if r(J) = eN for some idempotent e N, and if l(J) = Ng, for some
idempotent g N, then we say J is a left Baer-ideal.
In section 2, we see an example of right Baer-ideals which are not left Baer-ideal in a near field space N. We also see that in a near-field space N the set of Baer-ideals are closed under sum and direct product.
In section 3, I characterize near-field spaces over Baer-ideals in 2-by-2 generalized triangular matrix near-field spaces, full and upper triangular matrix near-field spaces. By these results I obtain new proofs for the well-known results about quasi-Baer and n-generalized quasi-Baer near-field spaces. Also, I obtained equivalent conditions for which the 2-by-2 generalized triangular matrix near-field space be right TA (i.e., for any two I; J N there is a K N such that r(I) + r(J) = r(K)).
In section 4, I prove that in a near-field space the product of two sub
near-field space over Baer ideals in a semi-prime near-field space S is a sub
near-field space over Baer-ideal. Also we show that an ideal J of a semi-prime
near-field space S is a near-field space over Baer-ideal if and only if int V (J) is
a clopen sub near-field space of Spec (S). Moreover, it is proved that an ideal J
of C(N) is a Baer-ideal if and only if int gJ Z(g) is a clopen sub near-field space
C of near-field space N.
In section 5, I derived main result on semi prime near-field space over
Baer-ideals.
Section 2: Preliminary results and examples Definition 2.1: An ideal J of near-field space N is called right Baer-ideal if
there exists an idempotent e N such that r(J) = eN, similarly, I can define left
Baer-ideal and we say J is a Baer-ideal if J is a right and left Baer-ideal. Example 2.2: The ideals 0 and N are Baer-ideals in any near-field space N. Example 2.3: For e Tr(N) the ideal NeN is a right Baer-ideal. Since, we have r(NeN) = r(eN) = r(Ne) = (1 e)N. Example 2.4: For g Tl(N), the ideal NgN is a left Baer-ideal. Since, l(NgN) = l(Ng) = l(gN) = N(1 g).
In the following, I provide an example of right Baer-ideals which are not left Baer-ideal. Also we can see a non-quasi-Baer near-field space which has a Baer-ideal.
Example 2.5: Let N =
2
2
0 Z
ZZ = {
b
an
0/ n Z, a, b Z2} where Z and Zn are
near-field space of integers and integers modulo n respectively.
(a) For ideal J =
2
2
0
0
Z
Z we have l(J) =
00
02Z and is not containing any
idempotent. Therefore J is not a left Baer-ideal of a near-field space N. on
the other hand r(J) =
00
2ZZ=
00
11N. Thus J is a right Baer-ideal of a
near-field space N.
(b) For ideal I =
00
02Z we have l(I) =
2
2
0
0
Z
Z = N
10
10 and r(I) =
2
2
0
0
Z
Z =
10
10N. Hence I is a Baer-ideal of a near-field space N.
Lemma 2.6: Let C1 and C2 be two right semi-central idempotents then (a) e1e2 is a right semi – central idempotent. (b) (e1 + e2 – e1e2) is a right semi – central
idempotent (c) T Tn(N) is finite then there is a is a right semi – central idempotent e such that NTN = NeN = < e >.
Proposition 2.7: The sum of two sub near-field spaces N1 and N2 over Baer-ideals in any near-field space N is a Baer-ideal. Proof: Let N1 and N2 be two sub near-field spaces over Baer-ideal of a near-
field space N. then there are idempotents e, g Tl(N) such that n(N1) = eN =
n(N(1 e )) and n(N2) = gN = n(N( 1 – g )). Therefore n(N1 + N2) = n(N1) n(N2) =
n(N( 1 – g )) = n(N( 1 – e ) + N( 1 – g )). Since, 1 – e, 1 – g Tn(N). by lemma 2.6,
we have k = (( 1 – e ) + ( 1 – g ) – ( 1 – e ) ( 1 – g ) ) Tn(N). on the other hand, we can see that n(N1 + N2) = n(N( 1 – e ) + N( 1 – g )) = n(Nk) = ( 1 – k)N. Hence N1 + N2 is a right near-field space over Baer-ideal. Similarly, we can see that N1 + N2 is a left near-field space over Baer-ideal. This completes the proof of the proposition.
Proposition 2.8: An ideal I of near-field space N = xXx N is a direct product
of near-field spaces is a right near-field space over Baer-ideal if and only if each
x(N) = Nx is a right near-field space over Baer-ideal Ix of N, where x : N Nx denote the canonical projection homomorphism. Proof: If I is a right sub near-field space over Baer-ideal of a near-field space N,
there exists an idempotent e N such that n(I) = eN. This implies that n( Ix ) =
x (e)Nx = exNx. Therefore each Ix is a right sub near-field space of Nx. Conversely, each Ix is a right sub near-field space over Baer-ideal, hence for
each x X there exists an idempotent ex Nx such that n(Ix) = exNx. Thus n(I) =
(ex)eX N. Therefore I is a sub near-field space over Baer-ideal of near-field space N. This completes the proof of the proposition.
Note 2.9: Let N = xX Nx, a direct product of near-field spaces. (a) N is quasi near-field space over Baer-ideal if and only if each Nx is quasi near-field space over Baer-ideal. (b) N is n-generalized quasi near-field space over Baer-ideal if and only if each Nx is n-generalized quasi near-field space over Baer-ideal.
Section 3: Sub near-field spaces over Baer-ideals in extension near-field spaces
In this section, S will denote a 2 2 generalized or formal triangular matrix
near-field space
2
1
0 N
MN , N2, N1 are near-field spaces and M is aff (N1, N2) –
Bi-module. If K is an (N1, N2) – bimodule of M briefly N1N N2 MN, then
AnnN (N) = { n N / Nn = 0 } and AnnT (N) = { t / tN = 0 }. In this section I
extended characterization of near-field spaces over Baer-ideal and 2 2 generalized triangular matrix near-field spaces by use a similar method as in Birkenmeier, kim and Park. Also I characterize near-field spaces over Baer-ideal in full and upper triangular matrix near-field spaces. By using of these results, I can prove the well-known results about quasi near-field spaces over Baer-ideal of near-field spaces N and generalized right quasi near-field spaces over quasi-Baer ideal of near-field spaces N. Theorem 3.1: An ideal I of Mn(N), I = Mn(J) for some right near-field space over Baer-ideal J of near-field space N. Proof: Let I be a right near-field space over Baer-ideal of Mn(N). I = Mn(J), for some ideal J of N. We claim that J is a right near-field space over Baer-ideal. By
hypothesis, F Ts(Mn(N))such that n(J) = FMn(N). Hence e11N n(J), where
e11 is the (1,1)-entries in F. We show that n(J) e11N. Suppose that y n(J)
and n(J) = Mn(n(J)). Hence B n(J), where b11 = y and zero elsewhere.
Therefore, B Mn(N). In matrix F of near-field space over Baer-ideal of Ts
(Mn(N)), eij = e11eij. This implies that y e11N. Now, let I = Mn(J) and J be a right
near-field space over Baer-ideal of Mn(N) in a near-field space N. Then an
idempotent e N n(J) = eN. So, n(Mn(J)) = Mn(n(J)) = Mn(eN) = FMn(N), where
In matrix F 1 i n, ea = e and eij = 0 i j. Thus I is a right near-field space over Baer-ideal of Mn(N). This completes the proof of the theorem.
Theorem 3.2: The following statements hold good. (a) J Tn (N), there are
ideals Ijk of N, 1 j, k n J =
nn
n
n
I
III
IIII
...000
.
.........
..0
...
22322
1131211
, Ijk Ijk+1 and Ij+1k Ijk.
(b) J is a right near-field space over Baer-ideal of near-field space Tn (N) if and only if each I1k is a right near-field space over Baer-ideal of N. ( c ) If M is a right near-field space over Baer-ideal of near-field space N, then Tn (M) is a right near-field space over Baer-ideal of near-field space Tn(N). Corollary 3.3: The following statements are hold good. (a) M is quasi near-field space over Baer-ideal of near-field space N if and only if Mn(M) is quasi near-field space over Baer-ideal of N. (b) M is n-generalized right quasi near-field space over Baer-ideal of N if and only if Mn(M) is n-generalized quasi near-field space over Baer-ideal of N. Proof: To prove (a) Let M be quasi near-field space over Baer-ideal of near-field
space N and I Mn(N). Then I = Mn(J) for some J N and J is a right near-
field space over Baer-ideal and so I is a right near-field space over Baer-ideal, hence Mn(N) is a quasi near-field space over Baer-ideal of near-field space N.
Now let J N and Mn(N) be quasi near-field space over Baer-ideal of near-field
space N. Then Mn(J) is a right near-field space over Baer-ideal of Mn(N) and again J is a right near-field space over Baer-idea in N, thus N is n-generalized right quasi -field space over Baer-ideal of a near-field space N.
To prove (b) Assume that J Mn(N) and N is a n-generalized right quasi near-
field space over Baer-ideal. Then I = Mn(J), where Jn is a right near-field space over Baer-ideal. This implies that In = Mn(Jn) is a right near-field space over Baer-ideal. This shows that Mn(N) is n-generalized quasi right near-field space over Baer-ideal. Converse is obvious. This completes the proof of the corollary. Corollary 3.4: M is quasi near-field space over Baer-ideal of N if and only if Mn(M) is quasi near-field space over Baer-ideal of N.
Proof: Let I Mn(J). By theorem 3.2, I =
nn
n
n
I
III
IIII
...000
.
.........
..0
...
22322
1131211
. By hypothesis,
Ijk is a right near-field space over a Baer-ideal. This implies that I is a right near-field space over a Baer-ideal. Thus Mn(N) is a quasi right near-field space over a Baer-ideal. Converse is obvious. This completes the proof of the theorem.
Note 3.5: Let e =
2
1
0 e
me be an idempotent element of S =
N
MT
0
(1) e Ts(S) if and only if (a) e1 Ts(T) (b) e2 Ts(N) (c) e1m = m and (d) e1me2 =
me2, m M.
(2) e1m = m if and only if eS
2
1
0
0
e
eS
(3) if e1me2 = me2 m M, then
2
1
0
0
e
eS eS.
(4) if e Ts(S), then
2
1
0
0
e
eS = eS.
Note 3.6: Let I =
Q
NJ
0 be an ideal of S =
N
KT
0. Then
r(I) =
)()(0
)()(
NAnnQr
JrJr
NN
NT and l(I) =
)(0
)()()(
Ql
QlNAnnIl
N
KTT.
Note 3.7: Let I =
Q
NJ
0 be an ideal of S =
N
KT
0. Then I is a right near-field
space over Baer-ideal of S if and only if (i) J is a right near-field space over
Baer-ideal of Tj (ii) rK(J) = (rT(J))K and (iii) rN(Q) AnnN(N) = aN, for some a2 = a
N.
Theorem 3.8: Let S =
N
KT
0. Then the following are equivalent. (i) S is quasi
near-field space over Baer-ideal of N. (ii) (a) N and T are quasi near-field spaces
over Baer-ideal of N. (b) rK(J) = (rT(J))K J Tj and (iii) If TNN T KN. then we
have AnnN(N) = aN for some a2 = a N.
Theorem 3.9: Let S =
N
KT
0. Then the following are equivalent. (i) S is n-
generalized right (principally) quasi near-field space over Baer-ideal of N. (ii) (a) If T is n-generalized right (principally) quasi near-field space over Baer-
ideal of N. (b) rK(Jn) = (rT(J) n)K J Tj and (c) If
I
NJ
0 T, some e2 = e
N rN(Jn) AnnN(Jn-1 N) AnnN(jn-2 NI) ……. AnnN(JN n-1) = eN. Proof: This theorem can be proved in cyclic method of proof.
To prove (i) (ii): (a), (b) let J T. then
OO
KJJ nn 1
is a near-field space
over Baer-ideal of S. It is clear that, Jn is a quasi near-field space over Baer-ideal of T, hence T is n-generalized right (principally) quasi near-field space over Baer-ideal and rK(Jn) = (rT(Jn))K.
( c ) If
I
NJ
0 S. then
n
nnnn
I
NINIJNJJ
0
...... 121
is a quasi near-field
space over Baer-ideal in S. Then some e for that e2 = e N such that
rN(Jn) AnnN(Jn-1 N) AnnN(jn-2 NI) ……. AnnN(JN n-1) = eN.
To prove that (ii) (i): Let Q =
I
NJ
0 T. by hypothesis and so
Qn =
n
nnnn
I
NINIJNJJ
0
...... 121
is a quasi near-field space over Baer-ideal
in S. hence S is n-generalized right (principally) quasi near-field space over Baer-ideal. Here we recall that a near-field space N is a right SA if for each J, I
N Q N such that r(J) + r(I) = r(Q). this completes the proof of the
theorem.
Theorem 3.10: Let S = =
N
KT
0. Then the following are equivalent.
(i) S is a right SA- near-field space over Baer-ideal of N.
(ii) (a) J1, J2 T, J3 T rK(J1) + rK(J2) = rK(J3), rT(J1) + rT(J2) = rT(J3) i.e.,
T is a right SA- near-field space over Baer-ideal of N and
(b) J, I N and (T, N) sub near-field spaces over Baer-ideals N1, N2 of K,
there are M N and TNN T KN rN(J) AnnN(N1) + rN(I) AnnN(N2) =
rN(M) AnnN(JN n-1). Proof: Clear and is obvious. Corollary 3.11: The following statements hold.
(a) Let N = T , J T, rK(J) = (rT(J))K. Then S is a right SA N is a right SA.
(b) Let N = T , K T, Then S is a right SA N is a right SA.
(c) Let T = K. Then S is right SA T is a right SA and J, I N and N1 , N2
T, there are M N and N T, such that rN(J) AnnN(N1) + rN(I) AnnN(N2) =
rN(M) AnnN(N).
Section 4: Semi prime near-field spaces over Baer-ideals
and near-field spaces of continuous functions
In this section, first, I show that an ideal J of C(N) is a near-field
space over Baer-ideal if and only if )(int fZJf is a clopen sub near-field
space over Baer-ideal of N. Then we show that an ideal J of semi-prime near-field space S is a near-field space over a Baer-ideal if and only if )(int JV is a
clopen sub near-field space of a near-field space Spec (S).
Also I prove that the product of two sub near-field spaces over a Baer-ideal in a semi-prime near-field space S is a near-field space over Baer-ideal.
Definition 4.1: A non-zero ideal J of S is an essential ideal if for any ideal I of S, 0IJ implies that I = 0.
Definition 4.2 [13]: An ideal P of a commutative near-field S is called pseudo-
prime ideal if ab = 0 implies that a P or b P.
We denote C(N) the near-field space of all real-valued continuous functions
on a completely regular Hausdorff space N.
Definition 4.3: For any f C( N ), Z( f ) = { y N|f(y) = 0 } is called a zero sub near-field space of near-field space N over Baer-ideal.
We can see that a sub near-field space B of near-field space S is clopen if
and only if B = Z( f ) for some idempotent f C(N). for any sub near-field space B of N we denote by Bint the interior of B that is the largest open sub near-
field space of a near-field space N contained in B over Baer-ideal.
Note 4.4: For J, I C(N), r( J ) = r( I ) )(int)(int gZfZ JfJf .
Proposition 4.5: The following statements hold. (a) An ideal J of C(N) is a near-
field spaces of all real valued continuous functions over a Baer-ideal
)(int fZJf is a clopen sub near-field space of a near-field space N. (b) Every
pseudo – prime ideal of C(N) is a near-field space over a Baer-ideal.
Corollary 4.6: C(N) is a Baer-near-field space N is an extremally
disconnected near-field space over a baer-ideal.
Proof: () Let W be a closed sub near-field space of N and C(N) is a near-field
space over a Baer ideal. By completely regularity of near-field space of N, there
exists an ideal J of C(N) such that W = )(gZJg . By proposition 4.5, Wint is
closed near-field space, hence N is extremally disconnected near-field space.
() Suppose that JC(N).Then, )(gZJg is closed near-field space. Obviously,
J is an ideal of a near-field space N over a Baer-ideal. Thus C(N) is a near-field space over a Baer-ideal. This completes the proof of the corollary.
Note 4.7 (a): b S, let supp(b) = { H Spec (S) : b H }. It is clear that by
(Theorem 3.1,[14B]) S, { supp(b) : b S } forms a basis of open near-field spaces on Spec (S). This topology is called hull-kernel topology. we mean of V(J)
is the sub near-field space of H Spec (S), where J H and V(J) = )(bVJb .
Note 4.7(b): It is true that (a) If J and I are two ideals of a semi-prime near-
field space S, then r( J ) = r( I ) )(int)(int IVJV . (b) B Spec (S) is a clopen
sub near-field space an idempotent e S such that B = V(e).
Proposition 4.8: Let S be a semi-prime near-field space over a Baer-ideal.
(a) An ideal J of S is a Baer-ideal )(int JV is a clopen sub near-field space of
Spec (S). (b) The product of two sub near-field spaces over Baer-ideals is a sub near-
field space over a Baer-ideal.
(c) If S is a commutative near-field space, then any essential sub near-field space over Baer-ideal of S is a sub near-field space over Baer-ideal.
Proof: Proof is obvious.
Section 5: Main Result on semi prime near-field space over
Baer-ideals
We apply the theory of near-field spaces over Baer-ideals to give the following renowned result.
Corollary 5.1: Let S be a semi-prime near-field space. Then S is a quasi
near-field space over Baer-ideal Spec (S) is extremally disconnected
near-field space over Baer-ideal.
Proof: Let B be a closed sub near-field space of Spec (S) and S is a
quasi sub near-field space over Baer-ideal. Since { V(b) | b S} is a base
for closed near-field spaces in Spec (S), Q S such that B = Qb V(b).
Take J = SQS. Then B = V(J). By [proposition 4.8], Bint is a closed near-
field space. Thus Spec (S) is extremally disconnected near-field space over a Baer-ideal.
Let J S, we know that V(J) is a closed sub near-field space of Spec (S). By
hypothesis and proposition 4.8, )(int JV is a clopen sub near-field space of
Spec (S), and hence J is Baer-ideal. Thus S is a quasi near-field space of a
near-field space over a Baer-ideal. This completes the proof of the corollary.
References
[1] A. Taherifar, Annihilator Conditions related to the quasi-Baer Condition, sub-mitted.
[2] A. Taherifar, Some new classes of topological spaces and annihilator ideals, Topol. Appl. 165 (2014), 84-97. [3] A. Pollinigher and A. Zaks, On Baer and quasi-Baer rings, Duke Math. J. 37 (1970),
pp.127-138.
[4] A. Moussavi, H. H. S. Javadi and E. Hashemi, Generalized quasi-Baer rings, Commun. Algebra 33 (2005), 2115-2129.
[5] F. Azarpanah, Essential ideals in C(X), Period. Math. Hungar. 31 (1995), 105-112.
[6] F. Azarpanah and O. A. S. Karamzadeh, Algebraic characterization of some disconnected spaces, Italian. J. Pure Appl. Math. 12 (2002), 155-168.
[7] G. F. Birkenmeier, Idempotents and completely semiprime ideals, Commun. Algebra, 11
( 1983), 567-580.
[8] G. F. Birkenmeier, M. Ghirati and A. Taherifar, When is a sum of annihilator ideals an annihilator ideal? Commun. Algebra (2014), accepted.
[9] G. F. Birkenmeier, H. E. Heatherly, J. Y. Kim and J. K. Park, Triangular matrix representations, Journal of Algebra 230 (2000), 558-595.
[10]G. F. Birkenmeier, J. Y. Kim and J. K. Park, A sheaf representation of quasi-Baer rings, Journal of Pure and Applied Algebra, 146 (2000), 209-223.
[11] G. F. Birkenmeier, J. Y. Kim , and J. K. Park, Quasi-Baer ring extensions and biregular rings, Bull. AUSTRAL. Math. Soc. 16 (2000), 39-52.
[12] G. F. Birkenmeier, J. Y. Kim and J. K. Park, Principally Quasi-Baer Rings, Commun. Algebra 29 (2001), 639-660.
[13] G. F. Birkenmeier, J. K. Park and S. T. Rizvi, Generalized triangular matrix rings and the fully invariant extending property, Rocky Mt. J. Math. 32 (4) (2002), 1299-1319.
[14A] Gillman and C. W. Khols, Convex and pseuodoprime ideals in rings of continuous functions, Math. Zeiteschr. 72 (1960), 399-409.
[14B] G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60.
[15] L. Gillman and C W Khols, convex pseudo prime ideals in rings of continuous functions,
Math zetscher, 72, 1960, pp. 399 – 409.
[16] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, 1976.
[17 N V Nagendram,T V Pradeep Kumar and Y V Reddy On “Semi Noetherian Regular Matrix
δ–Near Rings and their extensions”, International Journal of Advances in Algebra (IJAA),
Jordan, ISSN 0973-6964, Vol.4, No.1, (2011), pp.51-55.
[ 18 ] N V Nagendram,T V Pradeep Kumar and Y V Reddy “A Note on Bounded Matrices over a
Noetherian Regular Delta Near Rings”, (BMNR-delta-NR) published in International Journal
of Contemporary Mathematics, Vol.2, No.1, June 2011, Copyright@MindReaderPublications,
ISSNNo:0973-6298, pp.13-19.
[ 19 ] N V Nagendram,T V Pradeep Kumar and Y V Reddy “A Note on Boolean Regular Near-
Rings and Boolean Regular δ–Near Rings”, (BR-delta-NR) published in International Journal of
Contemporary Mathematics,IJCM Int. J. of Contemporary Mathematics ,Vol. 2, No. 1, June
2011,Copyright @ Mind Reader Publications, ISSN No: 0973-6298, pp. 29 - 34.
[ 20 ] N V Nagendram,T V Pradeep Kumar and Y V Reddy “on p-Regular δ–Near-Rings and
their extensions”, (PR-delta-NR) accepted and to be published in int. J. Contemporary
Mathematics (IJCM),0973-6298,vol.1, no.2, pp.81-85,June 2011.
[ 21 ] N V Nagendram, T V Pradeep Kumar and Y V Reddy “On Strongly Semi –Prime over
Noetherian Regular δ–Near Rings and their extensions”,(SSPNR-delta-NR) published in
International Journal of Contemporary Mathematics, Vol.2, No.1, June 2011, , pp.83-90.
[ 22 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On Structure Theory and
Planar of Noetherian Regular δ-Near–Rings (STPLNR-delta-NR)”,International Journal of
Contemporary Mathematics, IJCM ,published by IJSMA, pp.79-83, Dec, 2011.
[ 23 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On Matrix’s Maps over
Planar of Noetherian Regular δ-Near–Rings (MMPLNR-delta-NR)”,International Journal of
Contemporary Mathematics ,IJCM, published by IJSMA, pp.203-211, Dec, 2011.
[ 24 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On IFP Ideals on Noetherian
Regular-- Near Rings(IFPINR-delta-NR)”, Int. J. of Contemporary Mathematics, Copyright @
Mind Reader Publications, ISSN No: 0973-6298, Vol. 2, No. 1, pp.53-58, June 2011.
[ 25 ] N V Nagendram, B Ramesh paper "A Note on Asymptotic value of the Maximal size of a Graph
with rainbow connection number 2*(AVM-SGR-CN2*)" published in an International Journal of
Advances in Algebra(IJAA) Jordan @ Research India Publications, Rohini , New Delhi , ISSN 0973-
6964 Volume 5, Number 2 (2012), pp. 103-112.
[ 26 ] N V Nagendram and B Ramesh on "Polynomials over Euclidean Domain in Noetherian Regular
Delta Near Ring Some Problems related to Near Fields of Mappings(PED-NR-Delta-NR & SPR-NF)"
Accepted and published in an International Journal of Mathematical Archive (IJMA), An International
Peer Review Journal for Mathematical, Science & Computing Professionals ISSN 2229-
5046,vol.3,no.8,pp no. 2998-3002,2012.
[ 27 ] N V Nagendram research paper on "Near Left Almost Near-Fields (N-LA-NF)" communicated to
for 2nd intenational conference by International Journal of Mathematical Sciences and
Applications,IJMSA@mindreader publications, New Delhi on 23-04-2012 also for publication.
[ 28 ] N V Nagendram, T Radha Rani, Dr T V Pradeep Kumar and Dr Y V Reddy “A Generalized Near
Fields and (m, n) Bi-Ideals over Noetherian regular Delta-near rings (GNF-(m, n) BI-NR-delta-NR)",
published in an International Journal of Theoretical Mathematics and Applications
(TMA),Greece,Athens,dated 08-04-2012.
[ 29 ] N V Nagendram,Smt.T.Radha Rani, Dr T V Pradeep Kumar and Dr Y V Reddy "Applications of
Linear Programming on optimization of cool freezers(ALP-on-OCF)" Published in International Journal
of Pure and Applied Mathematics, IJPAM-2012-17-670 ISSN-1314-0744 Vol-75 No-3(2011).
[ 30 ] N V Nagendram "A Note on Algebra to spatial objects and Data Models(ASO-DM)" Published in
international Journal American Journal of Mathematics and Mathematical Sciences, AJMMS,USA,
Copyright @ Mind Reader Publications, Rohini , New Delhi, ISSN. 2250-3102, Vol.1, No.2 (Dec.
2012),pp. 233 – 247.
[ 31 ] N V Nagendram, Ch Padma, Dr T V Pradeep Kumar and Dr Y V Reddy "A Note on Pi-Regularity
and Pi-S-Unitality over Noetherian Regular Delta Near Rings(PI-R-PI-S-U-NR-Delta-NR)" Published in
International Electronic Journal of Pure and Applied Mathematics, IeJPAM-2012-17-669 ISSN-1314-
0744 Vol-75 No-4(2011).
[ 32 ] N V Nagendram, Ch Padma, Dr T V Pradeep Kumar and Dr Y V Reddy "Ideal Comparability
over Noetherian Regular Delta Near Rings(IC-NR-Delta-NR)" Published in International Journal of
Advances in Algebra(IJAA,Jordan),ISSN 0973-6964 Vol:5,NO:1(2012),pp.43-53@ Research India
publications,Rohini,New Delhi.
[ 33 ] N. V. Nagendram , S. Venu Madava Sarma and T. V. Pradeep Kumar , “A NOTE ON
SUFFICIENT CONDITION OF HAMILTONIAN PATH TO COMPLETE GRPHS (SC-HPCG)”,
IJMA-2(11), 2011 , pp.1-6.
[ 34 ] N V Nagendram,Dr T V Pradeep Kumar and Dr Y V Reddy “On Noetherian Regular Delta Near
Rings and their Extensions(NR-delta-NR)”,IJCMS,Bulgaria,IJCMS-5-8-2011,Vol.6,2011,No.6,255-262.
[ 35 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On Semi Noehterian Regular
Matrix Delta Near Rings and their Extensions(SNRM-delta-
NR)”,Jordan,@ResearchIndiaPublications,AdvancesinAlgebraISSN 0973-6964 Volume 4, Number 1
(2011), pp.51-55© Research India Publicationspp.51-55
[ 36 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On Boolean Noetherian Regular
Delta Near Ring(BNR-delta-NR)s”,International Journal of Contemporary Mathematics,IJCM Int. J. of
Contemporary Mathematics ,Vol. 2, No. 1-2,Jan-Dec 2011 , Mind Reader Publications,ISSN No: 0973-
6298,pp. 23-27.
[ 37 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On Bounded Matrix over a
Noetherian Regular Delta Near Rings(BMNR-delta-NR)”,Int. J. of Contemporary Mathematics,Vol. 2,
No. 1-2, Jan-Dec 2011 ,Copyright @ Mind Reader Publications, ISSN No: 0973-6298,pp.11-16
[ 38 ] N V Nagendram,Dr T V Pradeep Kumar and Dr Y V Reddy “On Strongly Semi Prime over
Noetherian Regular Delta Near Rings and their Extensions(SSPNR-delta-NR)”, Int. J. of Contemporary
Mathematics,Vol. 2, No. 1, Jan-Dec 2011 ,Copyright @ Mind Reader Publications ,ISSN No: 0973-
6298,pp.69-74.
[ 39 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On IFP Ideals on Noetherian
Regular Delta Near Rings(IFPINR-delta-NR)”, Int. J. of Contemporary Mathematics,Vol. 2, No. 1-2,
Jan-Dec 2011 ,Copyright @ Mind Reader Publications, ISSN No: 0973-6298,pp.43-46.
[ 40 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On Structure Thoery and Planar of
Noetherian Regular delta-Near–Rings (STPLNR-delta-NR)”,International Journal of Contemporary
Mathematics, IJCM ,accepted for international conference conducted by IJSMA, New Delhi December
18,2011,pp:79-83,Copyright @ Mind Reader Publications and to be published in the month of Jan 2011.
[ 41 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On Matrix’s Maps over Planar of
Noetherian Regular delta-Near–Rings (MMPLNR-delta-NR)”,International Journal of Contemporary
Mathematics ,IJCM, accepted for international conference conducted by IJSMA, New Delhi December
18,2011,pp:203-211,Copyright @ Mind Reader Publications and to be published in the month of Jan
2011.
[ 42 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “Some Fundamental Results on P-
Regular delta-Near–Rings and their extensions (PNR-delta-NR)”,International Journal of Contemporary
Mathematics ,IJCM,Jan-December’2011,Copyright@MindReader Publications,ISSN:0973-6298,
vol.2,No.1-2,PP.81-85.
[ 43 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “A Generalized ideal based-zero
divisor graphs of Noetherian regular Delta-near rings (GIBDNR- d-NR)" , International Journal of
Theoretical Mathematics and Applications (TMA)accepted and published by TMA, Greece,
Athens,ISSN:1792- 9687 (print),vol.1, no.1, 2011, 59-71, 1792-9709 (online),International Scientific
Press, 2011.
[ 44 ] N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy "Inversive Localization of
Noetherian regular Delta-near rings (ILNR- Delta-NR)" , International Journal of Pure And Applied
Mathematics published by IJPAM-2012-17-668, ISSN.1314-0744 vol-75 No-3,SOFIA,Bulgaria.
[ 45 ] N V Nagendram, S V M Sarma, Dr T V Pradeep Kumar "A note on Relations between Barnette
and Sparse Graphs" publishd in an International Journal of Mathematical Archive (IJMA), An
International Peer Review Journal for Mathematical, Science & Computing Professionals, 2(12),2011,
pg no.2538-2542,ISSN 2229 – 5046.
[ 46 ] N V Nagendram "On Semi Modules over Artinian Regular Delta Near Rings(S Modules-AR-
Delta-NR) Accepted and published in an International Journal of Mathematical Archive (IJMA)", An
International Peer Review Journal for Mathematical, Science & Computing Professionals ISSN 2229-
5046,IJMA-3-474,2012.
[ 47 ] N VNagendram1, N Chandra Sekhara Rao2 "Optical Near field Mapping of Plasmonic Nano
Prisms over Noetherian Regular Delta Near Fiedls (ONFMPN-NR-Delta-NR)" accepted for 2nd
international Conference by International Journal of Mathematical Sciences and Applications, IJMSA @
mind reader publications, New Delhi going to conduct on 15 – 16 th December 2012 also for
publication.
[ 48 ] N V Nagendram, K V S K Murthy(Yoga), "A Note on Present Trends on Yoga Apart From
Medicine Usage and Its Applications(PTYAFMUIA)" pubished by the International Association of
Journal of Yoga Therapy, IAYT 18 th August , 2012.
[ 49 ] N V Nagendram ,B Ramesh, Ch Padma , T Radha Rani and S V M Sarma research article "A Note
on Finite Pseudo Artinian Regular Delta Near Fields(FP AR-Delta-NF)" communicated to International
Journal of Advances in Algebra, IJAA ,Jordan on 22 nd August 2012.
[ 50 ] R. Engelking, General Topology, PWN-Polish Sci. Publ, 1977.
[ 51 ] S. K. Berberian, Baer -rings, Springer, Berlin, 1972.
[ 52 ] T. Y. Lam, A First Course in Non-Commutative Rings, New York, Springer, 1991.
[ 53 ] T. Y. Lam, Lecture on Modules and Rings, Springer, New York, 1999.
[ 54 ] V. Clark, Twisted matrix units semigroup algebra, Duke math. J. 34 (1967), 417-424.
