The union of two graphs associated with the set of all non-zero annihilating ideals of a commutative ring

Abstract

The rings considered in this article are commutative with identity which are not integral domains. Let R be a ring. Let A(R) denote the set of all annihilating ideals of R and let us denote A(R) \ {(0)} by A(R)^∗. With R, in this article we associate an undirected graph denoted by G(R) whose vertex set is A(R)^∗ and distinct vertices I and J are adjacent in G(R) if and only if either IJ ≠ (0) or I+J ∉ A(R). If R is reduced, then in Section 2, we prove that G(R) is connected and determine the diameter and radius of G(R). If R is not reduced, then in Section 3, we answer when G(R) is connected and determine the diameter and radius of G(R) when G(R) is connected.