Narrow Results

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.

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.

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.

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 $r\in R\backslash \{0\}$ 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 $A{(R)}^{\ast }$ 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 $\mathrm{\Omega }(R)$, which is an undirected simple graph whose vertex set is $A{(R)}^{\ast }$ and distinct elements $I,J\in A{(R)}^{\ast }$ are joined by an edge in this graph if and only if $I+J\in A(R)$. 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 ${(\mathrm{\Omega }(R))}^c$, where ${(\mathrm{\Omega }(R))}^c$ is the complement of $\mathrm{\Omega }(R)$. In this paper, we first determine when ${(\mathrm{\Omega }(R))}^c$ is connected and also determine its diameter when it is connected. We next discuss the girth of ${(\mathrm{\Omega }(R))}^c$ and study regarding the cliques of ${(\mathrm{\Omega }(R))}^c$. Moreover, it is shown that ${(\mathrm{\Omega }(R))}^c$ is complemented if and only if $R$ is reduced.

The rings considered in this paper are commutative with identity which are not integral domains. Let $R$ be a ring. Let us denote the set of all annihilating ideals of $R$ by $𝔸(R)$ and $𝔸(R)\setminus \{(0)\}$ by $𝔸{(R)}^{\ast }$. With $R$, we associate an undirected graph, denoted by $𝕊𝕊𝔸𝔾(R)$, whose vertex set is $𝔸{(R)}^{\ast }$ and distinct vertices $I$ and $J$ are adjacent in this graph if and only if $IJ=(0)$ and $I+J\in 𝔸(R)$. The aim of this paper is to study the interplay between the graph-theoretic properties of $𝕊𝕊𝔸𝔾(R)$ and the ring-theoretic properties of $R$.