Extended Annihilating-Ideal Graph of a Commutative Ring

Abstract

Abstract Let R be a commutative ring with identity. An ideal I of a ring R is called an annihilating-ideal if there exists a nonzero ideal J of R such that IJ = (0) and we use the notation 𝔸(R) for the set of all annihilating-ideals of R. In this paper, we introduce the extended annihilating-ideal graph of R, denoted by 𝔼𝔸𝔾(R). It is the simple graph with vertices 𝔸(R)* = 𝔸(R)\ {(0)}, and two distinct vertices I and J are adjacent whenever there exist two positive integers n and m such that InJm = (0) with In ≠ (0) and Jm ≠ (0). Here we discuss in detail the diameter and girth of 𝔼𝔸𝔾(R) and investigate the coincidence of 𝔼𝔸𝔾(R) with the annihilating-ideal graph 𝔸𝔾 (R). Moreover we propose open questions in this paper.