Narrow Results

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 $R$.

Let R = ℂ [x] be a polynomial ring with complex coefficients and D_X = ℂ〈x, ∂〉 be the Weyl algebra. Describing the localization R_f = R[f⁻¹] for nonzero f ∈ R as a D_X-module amounts to computing the annihilator A = Ann(f^a) ⊂ D_X of the cyclic generator f^a for a suitable negative integer a. We construct an iterative algorithm that uses truncated annihilators to build A for planar curves.