Perfectness of a graph associated with annihilating ideals of a ring

Abstract

Let [Formula: see text] be a commutative ring with identity which is not an integral domain. An ideal [Formula: see text] of a ring [Formula: see text] is called an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. Let [Formula: see text] be a simple undirect graph associated with [Formula: see text] whose vertex set is the set of all nonzero annihilating ideals of [Formula: see text] and two distinct vertices [Formula: see text] are joined if and only if [Formula: see text] is also an annihilating ideal of [Formula: see text]. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. In this paper, we characterize all rings whose [Formula: see text] is perfect.