AN EXTENSION OF ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS

Abstract

Let R be a commutative ring with unity. The extension of annihilating-ideal graph of R, AG(R), is the graph whose vertices are nonzero annihilating ideals of R and two distinct vertices I and J are adjacent if and only if there exist n,m ∈ N such that InJm = (0) with In, Jm 6= (0). First, we differentiate when AG(R) and AG(R) coincide. Then, we have characterized the diameter and the girth of AG(R) when R is a finite direct products of rings. Moreover, we show that AG(R) contains a cycle, if AG(R) 6= AG(R).