Author

# S. Visweswaran

Bio: S. Visweswaran is an academic researcher from Saurashtra University. The author has contributed to research in topics: Commutative ring & Jacobson radical. The author has an hindex of 5, co-authored 21 publications receiving 69 citations.

Topics: Commutative ring, Jacobson radical, Subring, Reduced ring, Minimal ideal

##### Papers

More filters

••

TL;DR: This paper considers a simple undirected graph associated with R denoted by Ω(R) whose vertex set equals the set of all nonzero annihilating ideals of R and two distinct vertices I, J in this graph are joined by an edge if and only if I + J is also an annihilating ideal of R.

Abstract: Let R be a commutative ring with identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R\{0} such that Ir = (0). In this paper, we consider a simple undirected graph associated with R denoted by Ω(R) whose vertex set equals the set of all nonzero annihilating ideals of R and two distinct vertices I, J in this graph are joined by an edge if and only if I + J is also an annihilating ideal of R. In this paper, for any ring R which is not an integral domain, the problem of when Ω(R) is connected is discussed and if Ω(R) is connected, then it is shown that diam(Ω(R)) ≤ 2. Moreover, it is verified that gr(Ω(R)) ∈ {3, ∞}. Furthermore, rings R such that ω(Ω(R)) < ∞ are characterized.

20 citations

••

TL;DR: In this paper, the relationship between the connectedness of the complement of the zero-divisor graph of a commutative ring with identity admitting at least two nonzero zero divisors is studied.

Abstract: Let R be a commutative ring with identity admitting at least two nonzero zero-divisors. First, in this article we determine when the complement of the zero-divisor graph of R is connected and also determine its diameter when it is connected. Second, in this article we study the relationship between the connectedness of the complement of the zero-divisor graph of R to that of the connectedness of the complement of the zero-divisor graph of T where either T = R[x] or T = R[[x]] and we study the relationship between their diameters in the case when both the graphs are connected. Finally, we give some examples to illustrate some of the results proved in this article.

17 citations

••

TL;DR: The aim of this paper is to study the interplay between the ring theoretic properties of a ring R and the graph theoretic Properties of (Ω(R))c, where (Χ(R) c) is the complement of Ω( R), and it is shown that ( Ω (R)c is complemented if and only if R is reduced.

Abstract: The rings considered in this paper are commutative with identity which are not integral domains. Recall that an ideal I of a ring R is called an annihilating ideal if there exists r ∈ R\{0} such that Ir = (0). As in [M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739], for any ring R, we denote by A(R) the set of all annihilating ideals of R and by A(R)∗ the set of all nonzero annihilating ideals of R. Let R be a ring. In [S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithm Appl. 6(4) (2014), Article ID: 1450047, 22pp], we introduced and studied the properties of a graph, denoted by Ω(R), which is an undirected simple graph whose vertex set is A(R)∗ and distinct elements I,J ∈ A(R)∗ are joined by an edge in this graph if and only if I + J ∈ A(R). The aim of this paper is to study the interplay between the ring theoretic properties of a ring R and the graph theoretic properties of (Ω(R))c, where (Ω(R))c is the complement of Ω(R). In this paper, we first determine when (Ω(R))c is connected and also determine its diameter when it is connected. We next discuss the girth of (Ω(R))c and study regarding the cliques of (Ω(R))c. Moreover, it is shown that (Ω(R))c is complemented if and only if R is reduced.

7 citations

••

27 Apr 2021TL;DR: In this article, the concept of S-primary ideal of a commutative ring was introduced and its properties were investigated. But the S-Laskerian rings considered in this paper are not commutativity with identity.

Abstract: The rings considered in this article are commutative with identity. This article is motivated by the results proved by Hamed and Malek (Beitr Algebra Geom 61:533–542, 2020) on S-prime ideals. In this paper, we introduce the concept of S-primary ideal of a commutative ring and study its properties. Let R be a ring and let S be a multiplicatively closed subset (m.c. subset) of R. Let $$\mathfrak {q}$$
be an ideal of R with $$\mathfrak {q}\cap S = \emptyset $$
. We say that $$\mathfrak {q}$$
is an S-primary ideal of R if there exists $$s\in S$$
such that for all $$a, b\in R$$
with $$ab\in \mathfrak {q}$$
, we have either $$sa\in \mathfrak {q}$$
or $$sb\in \sqrt{\mathfrak {q}}$$
. In Sect. 2, we investigate some basic properties of S-primary ideals of a commutative ring and in Sect. 3, we introduce the concept of S-Laskerian rings and study some of its properties.

7 citations

••

TL;DR: In this article, the authors considered a ring R with at least one nonzero annihilating ideal and determined necessary and sufficient conditions in order that the complement of its annihilating graph is connected and also finds its diameter when it is connected.

Abstract: Rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. For such a ring R, we determine necessary and sufficient conditions in order that the complement of its annihilating ideal graph is connected and also find its diameter when it is connected. We discuss the girth of the complement of the annihilating ideal graph of R and prove that it is either equal to 3 or ∞. We also present a necessary and sufficient condition for the complement of the annihilating ideal graph to be complemented.

6 citations

##### Cited by

More filters

••

TL;DR: This paper considers a simple undirected graph associated with R denoted by Ω(R) whose vertex set equals the set of all nonzero annihilating ideals of R and two distinct vertices I, J in this graph are joined by an edge if and only if I + J is also an annihilating ideal of R.

Abstract: Let R be a commutative ring with identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R\{0} such that Ir = (0). In this paper, we consider a simple undirected graph associated with R denoted by Ω(R) whose vertex set equals the set of all nonzero annihilating ideals of R and two distinct vertices I, J in this graph are joined by an edge if and only if I + J is also an annihilating ideal of R. In this paper, for any ring R which is not an integral domain, the problem of when Ω(R) is connected is discussed and if Ω(R) is connected, then it is shown that diam(Ω(R)) ≤ 2. Moreover, it is verified that gr(Ω(R)) ∈ {3, ∞}. Furthermore, rings R such that ω(Ω(R)) < ∞ are characterized.

20 citations

••

TL;DR: Graph Theory with Applications to Engineering and Computer as mentioned in this paper is an excellent introductory treatment of graph theory and its applications that has had a long life in the instruction of advanced undergraduates and graduate students in all areas that require knowledge of this subject.

Abstract: Graph Theory with Applications to Engineering and Computer ... This outstanding introductory treatment of graph theory and its applications has had a long life in the instruction of advanced undergraduates and graduate students in all areas that require knowledge of this subject. The first nine chapters constitute an excellent overall introduction, requiring only some knowledge of set theory and matrix algebra.

16 citations

••

TL;DR: In this paper, the complement of the zero-divisor graph with respect to a semiprime ideal I of a lattice L, is connected and also determined its diameter, radius, center center and girth.

Abstract: In this paper, we determine when \((\Gamma_I (L))^c\), the complement of the zero-divisor graph \((\Gamma_I (L))\) with respect to a semiprime ideal I of a lattice L, is connected and also determine its diameter, radius,
centre and girth. Further, a form of Beck's conjecture is proved for \((\Gamma_I (L))\) when \(\omega((\Gamma_I (L))c) 10.1017/S0004972713000300

15 citations