Some graph on prime ideals of a commutative ring

Abstract

Let R be a commutative ring with an identity. Let Spec(R) be the set of all prime ideals of R and Max(R) be the set of all maximal ideals of R. Let S  Max(R). We de ne an S-proper ideal sum graph on Spec(R), denoted by 􀀀S(Spec(R); S), as an undirected graph whose vertex set is the set Spec(R) and, for two distinct vertices P and Q, there is an arc from P to Q, whenever P +Q M, for some maximal idealMin S. In this paper, we prove that the complement graph of a proper sum graph 􀀀(Spec(R); S) is complete if and only if R is an Artinian ring. We also study some basic properties of the graph 􀀀S(Spec(R); S) such as connectivity, girth and clique number. We explore the in uence of the ring theoretic properties of a commutative ring R on the proper sum graph of R and vice versa.