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INTERNATIONAL JOURNAL OF CONTEMPORARY ADVANCD MATHEMATICS

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VOLUME 1, ISSUE 2, 2011

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DR. NABEEL TAHIR

ISSN (Online): 2180 - 1266

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EDITORIAL PREFACE

The International Journal of Contemporary Advanced Mathematics (IJCM) is an effective medium for interchange of high quality theoretical and applied research in Computational Linguistics from theoretical research to application development. This is the Second Issue of Volume One of IJCM. The Journal is published bi-monthly, with papers being peer reviewed to high international standards. International Journal of Contemporary Advanced Mathematics (IJCM) publish papers that describe state of the art techniques, scientific research studies and results in computational linguistics in general but on theoretical linguistics, psycholinguistics, natural language processing, grammatical inference, machine learning and cognitive science computational models of linguistic theorizing: standard and enriched context free models, principles and parameters models, optimality theory and researchers working within the minimalist program, and other approaches. IJCM give an opportunity to scientists, researchers, and vendors from different disciplines of Artificial Intelligence to share the ideas, identify problems, investigate relevant issues, share common interests, explore new approaches, and initiate possible collaborative research and system development. This journal is helpful for the researchers and R&D engineers, scientists all those persons who are involve in Contemporary Advanced Mathematics. Highly professional scholars give their efforts, valuable time, expertise and motivation to IJCM as Editorial board members. All submissions are evaluated by the International Editorial Board. The International Editorial Board ensures that significant developments in image processing from around the world are reflected in the IJCM publications. IJCM editors understand that how much it is important for authors and researchers to have their work published with a minimum delay after submission of their papers. They also strongly believe that the direct communication between the editors and authors are important for the welfare, quality and wellbeing of the Journal and its readers. Therefore, all activities from paper submission to paper publication are controlled through electronic systems that include electronic submission, editorial panel and review system that ensures rapid decision with least delays in the publication processes. To build its international reputation, we are disseminating the publication information through Google Books, Google Scholar, Directory of Open Access Journals (DOAJ), Open J Gate, ScientificCommons, Scribd, CiteSeerX Docstoc and many more. Our International Editors are working on establishing ISI listing and a good impact factor for IJCM. We would like to remind you that the success of our journal depends directly on the number of quality articles submitted for review. Accordingly, we would like to request your participation by submitting quality manuscripts for review and encouraging your colleagues to submit quality manuscripts for review. One of the great benefits we can provide to our prospective authors is the mentoring nature of our review process. IJCM provides authors with high quality, helpful reviews that are shaped to assist authors in improving their manuscripts.

Editorial Board Members International Journal of Contemporary Advanced Mathematics (IJCM)

EDITORIAL BOARD

Editor-in-Chief (EiC) Professor En-Bing Lin

Central Michigan University (United States of America)

ASSOCIATE EDITORS (AEiCS)

Dr. Yang Wang Michigan State University United States of America

EDITORIAL BOARD MEMBERS (EBMs)

Dr. Armen G. Bagdasaryan V.A. Trapeznikov Institute for Control Sciences Russia Dr. Taher Abualrub American University of Sharjah United Arab Emirates

International Journal of Contemporary Advanced Mathematics (IJCM), Volume (1) : Issue (2) : 2011

Table of Content

Volume 1, Issue 2, January / February 2011

Pages

16 - 22

On the Dimension of the Quotient Ring R/K Where K is a Complement

Satyanarayana Bhavanari, Nagaraju Dasari, Babu Prasad Munagala, Mohiddin

Shaw Shaik

Bh. Satyanarayana, D. Nagaraju, M. Babu Prasad & Sk. Mohiddin Shaw

International Journal of Contemporary Advanced Mathematics, (IJCM), Volume (1): Issue (2) 16

On the Dimension of the Quotient Ring R/K Where K is a Complement

Satyanarayana Bhavanari [email protected] Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar-522 510, AP, India.

Nagaraju Dasari [email protected] Department of Science & Humanities (Mathematics Division), HITS, Hindustan University, OMR, Padur, Chennai – 603 103, India.

Babu Prasad Munagala Vijayawada, Andhra Pradesh, India.

Mohiddin Shaw Shaik [email protected] Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar-522 510, AP, India.

Abstract

The aim of the present paper is to obtain some interesting results related to the concept “finite dimension” in the theory of associative rings R with respect to two sided ideals. It is known that if an ideal H of R has finite dimension, then there

exist uniform ideals Ui, 1 ≤ i ≤ n of R such that the sum U1 ⊕ U2 ⊕ … ⊕ Un is essential in H. This n is independent of choice of uniform ideals and we call it as dimension of H (we write dim H, in short). We obtain some important relations between the concepts complement ideals and essential ideals. Finally, we proved that dim(R/K) = dim R – dim K for a complement ideal K of R. We include some necessary examples. Keywords: Ring, Two Sided Ideal, Essential Ideal, Uniform Ideal, Finite Dimension, Complement Ideal.

1. INTRODUCTION

The dimension of a vector space is defined as the number of elements in the basis. One can define a basis of a vector space as a maximal set of linearly independent vectors or a minimal set of vectors which span the space. The former when generalized to modules over rings becomes the concept of Goldie dimension. Goldie proved a structure theorem for modules which states that “a module with finite Goldie dimension (FGD, in short) contains a finite number of uniform submodules U1, U2, …, Un whose sum is direct and essential in M”. The number n obtained here is independent of the choice of U1, U2, …, Un and it is called as Goldie dimension of M. The concept Goldie dimension in Modules was studied by several authors like Satyanarayana, Mohiddin Shaw.



If we consider ring as a module over itself, then the existing literature tells about dimension theory for ideals (i.e., two sided ideals) in case of commutative rings; and left (or right) ideals in case of associative (but not commutative) rings. So we can understand the structure theorem for associative rings in terms of one sided ideals only (that is, if R has FGD with respect to left (right) ideals, then there exist n uniform left (or right) ideals of R whose sum is direct and essential in R). This result cannot say about the structure theorem for associative rings in terms of two sided ideals.

To fill this gap, Satyanarayana, Nagaraju, Balamurugan & Godloza [4] started studying the concepts: complement, essential, uniform, finite dimension with respect to two sided ideals of R. We say a ring R has finite dimension on ideals (FDI, in short) if R does not contain an infinite number of non-zero ideals of R whose sum is direct. A non- zero ideal K of R is said to have finite dimension on ideals of R (FDIR, in short) if K does not contain an infinite number of non-zero ideals of R whose sum is direct. It is clear that if R has FDI, then every non-zero ideal of R has FDIR.

Now we state some definitions and results from [4 & 5] that are useful in the later part of this

paper. We write “I ⊴ R” to denote “I is an ideal (two sided ideal) of R”.

1.1 Definitions: Let I ⊴ R, J ⊴ R such that I ⊆ J.

(i). We say that I is essential (or ideal essential) in J if it satisfies the following condition: K ⊴ R,

K ⊆ J, I ∩ K = (0) imply K = (0). (ii). If I is essential in J and I ≠ J, then we say that J is a proper essential extension of I. If I is

essential in J, then we denote this fact by I ≤e J.

(iii). If K ⊴ R, A ⊴ R and K is a maximal element in {I / I ⊴ R, I ∩ A = (0)}, then we say that K is a complement of A (or a complement in R). 1.2 Note: If A, B, C are ideals of R, A ⊆ C, A ∩ B = (0) and C is a complement of B, then

C ⊕ B ≤e R, and C is an essential extension of A.

1.3 Result (2.4 of [4]): (i) If I ⊴ R, J⊴ R, K⊴ R such that I ≤e J, and J ≤e K, then I ≤e K;

(ii) If I ⊆ J ⊆ K, then I ≤e K if and only if I ≤e J, and J ≤e K; and

(iii) If R, S are two rings, f: R → S is a ring isomorphism, and A is an ideal of R, then A ≤e R ⇔

f(A) ≤e S. 1.4 Lemma (2.7 of [4]): Let K1, K2, … Kt, L1, L2, … Lt are ideals of R such that the sum

K1 + K2 + … + Kt is direct and Li ⊆ Ki for 1 ≤ i ≤ t. Then L1 + L2 + … + Lt ≤e K1 + K2 + … + Kt

⇔ Li ≤e Ki for 1 ≤ i ≤ t.

1.5 Definition: A non-zero ideal I of R is said to be uniform if (0) ≠ J ⊴ R, and J ⊆ I ⇒ J ≤e I.

1.6 Theorem (3.3 of [4]): (i) I is an uniform ideal ⇔ L ⊴ R, K ⊴ R, L ⊆ I, K ⊆ I, L ∩ K = (0) ⇒ L = (0) or K = (0).

(ii) Let R and S be two rings and f: R → S be ring isomorphism. If U ⊴ R, then U is uniform in

R ⇔ f(U) is uniform in S.

(iii) If U and K are two ideals of R such that U ∩ K = (0), then U is uniform in R ⇔ (U + K)/K is uniform in R/K.

(iv). If R has FDI and (0) ≠ K ⊴ R, then K contains an uniform ideal of R. Now we state the main theorem on [4].



1.7 Theorem (4.4 of [4]): Suppose 0 ≠ H ⊴ R and H has FDIR. Then the following conditions hold. (i) (Existence) There exist uniform ideals U1, U2, … Un of R whose sum is direct and essential in H;

(ii) (Uniqueness) If Vi, 1 ≤ i ≤ k are uniform ideals of R whose sum is direct and essential in H, then k = n.

The number is independent of the choice of the uniform ideals Ui, 1 ≤ i ≤ n. This number n is called the dimension of H and it is denoted by dim H. 1.8 Theorem (2.2 of [5]): Suppose R has FDI.

(i). If H ⊴ R, K ⊴ R and H ⊆ K, then dim H ≤ dim K;

(ii) If (0) ≠ Ai is an ideal of R for all i, 1 ≤ i ≤ t whose sum is direct, and Ai ⊆ H, 1 ≤ i ≤ t, then

dim H ≥ t;

(iii) H is uniform ⇔ dim H = 1;

(iv) If H is a non-zero ideal of R, then dim H ≥ 1;

(v) If Ii, 1 ≤ i ≤ k are uniform ideals of R whose sum is direct, then k ≤ dim R. Moreover

dim H = max{k / there exist uniform ideals Ii, 1 ≤ i ≤ k of R whose sum is direct, Ii ⊆ H, 1 ≤ i ≤ k}; (vi). If n = dim R, then the number of summands in any decomposition of a given ideal I of R as a direct sum of non-zero ideals of R is at most n.; and

(vii) If f: R → S is an isomorphism and R has FDI, then S has FDI and dim R = dim S.

1.9 Result (2.3 of [5]): If H and K are ideals of R with H ∩ K = (0), then dim (K + H) = dim K + dim H.

1.10 Theorem (3.1 of [5]): If R has FDI with dim R = n and H ⊴ R, then the following conditions are equivalent:

(i). H ≤e R; (ii). dim H = dim R; and (iii). H contains a direct sum of n uniform ideals.

1.11 Proposition (3.4 of [5]): Suppose R has FDI and K ⊴ R.

(i) K is a complement ideal ⇔ K has no proper essential extensions; and

(ii) If K is a complement, then R/K has FDI, and dim(R/K) ≤ dim R. The aim of the present paper is to continue the study of rings with FDI. Section-2, deals with the concepts: Complement and Essential Ideals. In Section-3, we include an example of an ideal K

of R with dim R/K ≠ dim R – dim K. Finally, we proved that if K is a complement ideal of R, then dim R/K = dim R – dim K. Throughout this paper R stands for a fixed (not necessarily commutative) ring with FDI.

2. COMPLEMENT AND ESSENTIAL IDEALS 2.1 Lemma: Let K be an ideal of R and π: R → R/K be the canonical epimorphism. Then the following two conditions are equivalent: (i) K is a complement; and (ii) For any ideal K

1 of R containing K, we have that K

1 is a complement in R if and only if

π(K1) is complement in R/K.

Proof: (i) ⇒ (ii): Suppose that K is a complement of an ideal Z of R. Suppose K1 is a

complement ideal of R containing K. Now K1 is a complement of some ideal S of R. To

show π(K1) is a complement of π(S), it is enough to verify that π(K

1) is maximal with respect to

the property π(K1) ∩ π(S) = (0).



Let x ∈ π(K1) ∩ π(S) ⇒ x ∈ π(K

1) and x ∈ π(S) ⇒ x = k

1 + K and x = s + K, for some k

1 ∈ K

1

and s ∈ S ⇒ s - k1 ∈ K ⊆ K

1 ⇒ s ∈ K

1 (since k

1 ∈ K

1) ⇒ s ∈ K

1 ∩ S (since s ∈ S) ⇒ s = 0

(since K1

∩ S = (0)) ⇒ x = s + K = 0. Therefore π(K1) ∩ π(S) = (0).

Let A be an ideal of R/K such that A ⊋ π(K1). It is obvious that A = π(K

*) for some ideal K

* of

R with K* ⊇ K

1. If K

* = K

1, then A = π(K

*) = π(K

1), a contradiction. So K

* ⊋ K

1. Since K

1 is a

complement of S, we have that K* ∩ S ≠ (0). Let 0 ≠ y ∈ K

* ∩ S. Now y + K ∈ π(K

*) ∩ π(S). If

y + K = 0, then y ∈ K ⇒ y ∈ K ∩ S ⊆ K*

∩ S = (0), a contradiction. Hence

0 ≠ y + K ∈ π(K*) ∩ π(S). This shows that π(K

1) is a complement of π(S).

Conversely suppose that π(K1) = K

1/K is a complement of an ideal π(I) = I/K of R/K. Now we

have to verify that K1 is a complement ideal in R. By Note 1.2, there exists a complement X of

K such that Z ⊆ X. Since K1

∩ I = K, we have that K1

∩ (I ∩ Z) = (K1

∩ I) ∩ Z = K ∩ Z = (0) and

so K1

∩ (I ∩ Z) = (0). Let Y be a complement of I ∩ Z with Y ⊇ K1. Now since Y ⊇ K

1 ⊇ K and

I ⊇ K, we have Y ∩ I ⊇ K. Also (Y ∩ I) ∩ Z = Y ∩ (I ∩ Z) = (0). Since Y ∩ I ⊇ K, (Y ∩ I) ∩ Z =

(0) and K is a complement of Z, it follows that Y ∩ I = K. So π(Y) ∩ π(I) = (0). Since Y ⊇ K1,

we have π(Y) ⊇ π(K1). Now π(Y) ⊇ π(K

1), π(Y) ∩ π(I) = (0) and π(K

1) is a complement of π(I),

it follows that π(Y) = π(K1). Now we have that Y = K

1. [Verification: We know that Y ⊇ K

1. Let

x ∈ Y. Then π(x) ∈ π(Y) = π(K1) ⇒ x + K ∈ π(K

1) ⇒ x + K = y + K, for some y ∈ K

1

⇒ x - y ∈ K ⊆ K1 and y ∈ K

1 ⇒ x - y ∈ K

1 and y ∈ K

1 ⇒ x ∈ K

1. Therefore Y = K

1]. Since Y

is a complement, we conclude that K1 is a complement.

(ii) ⇒ (i): Since K is an ideal of R containing K, and since π(K) = 0 is a complement in R/K, it follows that K is complement in R.

2.2 Lemma: Let K ⊴ R and π: R → R/K be the canonical epimorphism. Then the following two conditions are equivalent: (i) K is a complement; and

(ii) For any essential ideal S of R, π(S) is essential in R/K.

Proof: (i) ⇒ (ii): Let S be an essential ideal of R. To show π(S) is essential in R/K, take an

ideal Z/K of R/K such that π(S) ∩ (Z/K) = (0). It is enough to show Z = K. In a contrary way,

suppose Z ≠ K. Then by Proposition 1.11 (i), K has no proper essential extensions. So K is not

essential in Z and hence there exists an ideal (0) ≠ A of R such that A ∩ K = (0) and A ⊆ Z.

Since S is essential in R, there exists 0 ≠ x ∈ S ∩ A ⇒ π(x) ∈ π(S) ∩ π(A) ⊆ π(S) ∩ π(Z) = (0)

⇒ π(x) = 0 ⇒ x + K = 0 ⇒ x ∈ K ⇒ x ∈ K ∩ A (since x ∈ A) = (0) ⇒ x = 0, a contradiction.

Thus Z = K. We proved that π(S) is essential in R/K.

(ii) ⇒ (i): Assume the converse hypothesis. In a contrary way, suppose that K is not a complement. By Proposition 1.11 (i), K has a proper essential extension K

*. Let X be a

complement of K* in R. Then K

* ⊕ X is essential in R (by Note 1.2). Since K is essential in

K* by Lemma 1.4, K ⊕ X is essential in K

* ⊕ X and so K ⊕ X is essential in R (by Result 1.3 (i)).

By the converse hypothesis, we get that π(K + X) is essential in R/K. Since K* contains K

properly, π(K*) is a non-zero ideal of R/K. Now (K + X) ∩ K

* = K + (X ∩ K

*) = K, which shows

that π(K + X) ∩ π(K*) = (0). This is a contradiction to the fact that π(K + X) is essential in R/K.

Combining Lemmas 2.1 and 2.2, we get the following Theorem.

2.3 Theorem: Let K be an ideal of R and π: R → R/K be the canonical epimorphism. Then the following three conditions are equivalent: (i) K is a complement; (ii) For any ideal K

1 of R containing K, we have that K

1 is a complement in R if and only if

π(K1) is complement in R/K; and

(iii) For any essential ideal S of R, π(S) is essential in R/K.



3. DIMENSION OF THE QUOTIENT RING R/K 3.1 Lemma: Let R be a Ring with FDI. If A is an ideal of R such that dim(R/A) = 1 and A is not essential in R, then dim(R/A) = dim R - dim A.

Proof: Since A is not essential, there is a non-zero ideal I of R such that A ∩ I = (0). Let K be

a complement of A containing I. Suppose dim K ≥ 2. Then K contains a direct sum of two

uniform ideals I1 and I2 of R. Clearly Ii ∩ A = (0) for i = 1, 2. By Theorem 1.6 (iii),

1I+AA

,

2I +AA

are two uniform ideals of R/A. It is easy to verify that the sum

1I+AA

+

2I +AA

is

direct and hence dim(R/A) ≥ 2, a contradiction. Hence dim K ≱ 2. Since K ≠ (0), by Theorem

1.8 (iv), we have that dim K ≥ 1. Therefore dim K = 1. Since K is complement of A, the sum K + A is direct and essential in R. So dim R = dim(K + A) (by Theorem 1.10) = dim K + dim A (by Result 1.9) = 1 + dim A = dim (R/A) + dim A. Hence dim(R/A) = dim R - dim A.

It is well known that if V is a finite dimensional vector space and W is a subspace of V, then dim(V/W) = dim V - dim W. This dimension condition may not hold for a general ideal W of a Ring V where “dim” denotes the “finite dimension”. For this, observe the following examples.

3.2 Examples: Write R = ℤ, the ring of integers. Since every ideal of ℤ is essential in ℤ, it

follows that ℤ is uniform and so dim R = 1.

(i) Write K = 6ℤ. Now K is an uniform ideal of R. So dim K = 1 and dim R - dim K = 1 - 1 = 0.

Now R/K = ℤ/6ℤ ≅ ℤ6 ≅ ℤ2 + ℤ3 and so dim(R/K) = 2.

Thus dim(R/K) = 2 ≠ 0 = dim R - dim K.

(ii) Let p, q be distinct primes and consider H, the ideal of ℤ generated by the product of these

primes (that is, H = pqℤ). Now H is uniform ideal and so dim H = 1. It is known that

ℤ/H = ℤpq ≅ ℤp ⊕ ℤq, and ℤp, ℤq are uniform ideals. So dim(ℤ/H) = 2. Thus dim (ℤ/H) = 2 ≠ 0 =

1 - 1 = dim ℤ - dim H.

Hence, there arise a type of ideals K which satisfy the condition dim(R/K) = dim R – dim K.

3.3 Theorem: If R has FDI and K is a complement ideal, then dim(R/K) = dim R – dim K.

Proof: By Proposition 1.11 (ii), we have that R/K has FDI. If dim(R/K) = 1, then by Lemma 3.1,

dim(R/K) = dim R – dim K. Suppose dim(R/K) = m, where m ≥ 2. Then by Theorem 1.7, there exist ideals K1, K2, …, Km of R containing K properly such that Ki/K is an uniform ideal for

1 ≤ i ≤ m, the sum (K1/K) + (K2/K) + … + (Km/K) is direct and essential in R/K. Clearly K = Ki ∩ Kj,

i ≠ j for 1 ≤ i ≤ m and 1 ≤ j ≤ m. Since K is a complement ideal of R, by Proposition 1.11 (i), we

have that K is not essential in Ki, 1 ≤ i ≤ m. So there exist uniform ideals Ii (1 ≤ i ≤ m) of R such

that Ii ⊆ Ki and Ii ∩ K = (0). By a straight forward verification, we get that the sum K + I1 + I2 + … + Im is direct. Now we verify that T = K + I1 + I2 + … + Im is essential in R. Let H be an ideal of R

such that T ∩ H = (0). Then T ∩ (H + K) = (T ∩ H) + K (by modular law) = (0) + K = K. So (T/K)

∩ ( )H+K

K = (0). Since

( )iI+KK

is a non-zero ideal of the uniform ideal Ki/K, it follows that

( )iI+KK

is essential in Ki/K, for 1 ≤ i ≤ m. By Lemma 1.4, ( )1I+KK

+ ( )2I +K

K + ... +

( )mI +KK

is

essential in R/K. Therefore T/K is essential in R/K. Since (T/K) ∩ ( )H+K

K = (0), we have



( )H+KK

= (0) and so H ⊆ K. So H = H ∩ K ⊆ H ∩ T = (0). This shows that the sum T = K + I1 +

… + Im is essential in R. Now dim R = dim(K + I1 + … + Im) (by Theorem 1.10) =

dim K + dim I1 + dim I2 + … + Im (by Result 1.9) = 14243m-terms

dimK+1+...+1 (by Theorem 1.8 (iii)) = dim K

+ m = dim K + dim(R/K). Therefore dim(R/K) = dim R – dim K.

4. COMPARISON WITH THE PREVIOUS WORK DONE IN THE RELEVANT FIELDS

(i) A module is a generalized concept of vector space. If W is a subspace of a vector space V and dim W = dim V, then V = W. But in case of modules, if W is a submodule of a module M with dim W = dim M, then W is essential in M, but W may not be equal to M. Due to this fact the study of Goldie dimension in modules becomes important. A ring is a module over itself. So the theory developed in modules is also a contribution to the theory of Rings.

(ii) A ring R is a module RR. The right ideals in R coincide with the submodules of RR. So the dimension theory developed in modules speaks about the results related to the dimension of one sided “right ideals” of rings. But the results obtained in module theory can not speak about the dimension of two sided ideals of rings. To fill this gap, Satyanarayana, Nagaraju, Bala Murugan, and Godloza [4] started studying the concept ‘dimension of two sided ideals’ in rings. The study was continued in [5] and [7]. The further study on this concept formed the results of the present paper.’

5. CONCLUSION & FUTURE WORK This paper is the continuation of the published papers [4] and [5]. In this present paper, we are able to obtain several interesting results related to the concept dimension of rings with respect to two sided ideals. We proved fundamental and critical relations between complement ideals and essential ideals. In general the statement: dim(R/K) = dim R – dim K, is not true for two sided ideals K. To explain this fact, an example was presented. Finally we achieved the result and able to prove the important statement that dim(R/K) = dim R – dim K, for a particular type of submodule (namely, complement submodule). We continue this work, in near future, to get some more important dimension conditions in rings with respect to two sided ideals.

6. ACKNOWLEDGEMENT The first and fourth authors are thankful to the UGC (New Delhi) for assistance under the grant No. F.34-136/2008 (SR), dt. 30 DEC 2008. First author also thank the Walter Sisulu University, South Africa for inviting him as a Visiting Professor (March 26

th - April 10

th, 2007) during which period a part of the paper

was initiated. The second author acknowledges the authorities of Hindustan University for their encouragement. The authors acknowledges the referees for their valuable comments for the improvement of the paper.

7. REFERENCES

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2. Bh. Satyanarayana “A note on E-direct and S-inverse Systems”, Proc. of the Japan

Academy, 64-A:292-295, 1988. 3. Bh. Satyanarayana and Sk. Mohiddin Shaw “Fuzzy Dimension of Modules over Rings

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4. Bh. Satyanarayana D. Nagaraju, K. S. Balamurugan and L. Godloza “Finite Dimension in Associative Rings”, Kyungpook Mathematical Journal, 48:37-43, 2008.



5. Bh. Satyanarayna, D. Nagaraju, L. Godloza and S. Sreenadh “Some Dimension Conditions in Rings with Finite Dimension”, The PMU Journal of Humanities and Sciences 1(1):69-75, 2010.

6. Bh. Satyanarayana, D. Nagaraju, Sk. Mohiddin Shaw and S. Eswaraiah Setty “E-irreducible

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Rings”, East Asian Mathematical Journal 24:377-387, 2008.

8. Bh. Satyanarayana and K. Syam Prasad “Discrete Mathematics and Graph Theory” Prentice Hall of India, New Delhi, (2009) (ISBN: 978-81-203-3842-5).

INSTRUCTIONS TO CONTRIBUTORS The International Journal of Contemporary Advanced Mathematics (IJCM) brings together both of these aspects of biology and creates a platform for exploration and progress of these, relatively new disciplines by facilitating the exchange of information in the fields of computing and statistics, mathematics in the Technology sciences are expected to have a substantial impact on the scientific, engineering and economic development of the world. Together they are a comprehensive application of mathematics, statistics, science and computer science with an aim to understand living systems. We invite specialists, researchers and scientists from the fields of computer science, mathematics, statistics, physics and such related sciences to share their understanding and contributions towards scientific applications that set scientific or policy objectives, motivate method development and demonstrate the operation of new methods in the field of Contemporary Advanced Mathematics. To build its International reputation, we are disseminating the publication information through Google Books, Google Scholar, Directory of Open Access Journals (DOAJ), Open J Gate, ScientificCommons, Docstoc and many more. Our International Editors are working on establishing ISI listing and a good impact factor for IJCM. The initial efforts helped to shape the editorial policy and to sharpen the focus of the journal. Started with Volume 1, 2010, IJCM appears in more focused issues. Besides normal publications, IJCM intend to organized special issues on more focused topics. Each special issue will have a designated editor (editors) – either member of the editorial board or another recognized specialist in the respective field. We are open to contributions, proposals for any topic as well as for editors and reviewers. We understand that it is through the effort of volunteers that CSC Journals continues to grow and flourish.

LIST OF TOPICS The realm of International Journal of Contemporary Advanced Mathematics extends, but not limited, to the following:

• Biomedical modelling and computer simulation

• Computational genomics

• Computational intelligence • Computational proteomics

• DNA assembly, clustering, and mapping • E-health

• Fuzzy logic • Gene expression and microarrays

• Gene identification and annotation • Genetic algorithms

• Hidden Markov models • High performance computing

• Molecular evolution and phylogeny • Molecular modelling and simulation

• Molecular sequence analysis • Neural networks

CALL FOR PAPERS Volume: 2 - Issue: 1 i. Paper Submission: April 30, 2011 ii. Author Notification: May 31, 2011

iii. Issue Publication: June 2011

CONTACT INFORMATION

Computer Science Journals Sdn BhD

M-3-19 Plaza Damas, Sri Hartamas 50480, Kuala Lumpur

Malaysia

Phone: 006 03 6207 1607 006 03 2782 6991

Fax: 006 03 6207 1697

Email: [email protected]

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