SOME GRAPH ON PRIME IDEALS OF A COMMUTATIVE RING

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 define an S-proper ideal sum graph on Spec(R), denoted by ΓS(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 ideal M in S. In this paper, we prove that the complement graph of a proper sum graph Γ(Spec(R), S) is complete if and only if R is an Artinian ring. We also study some basic properties of the graph ΓS(Spec(R), S) such as connectivity, girth and clique number. We explore the influence of the ring theoretic properties of a commutative ring R on the proper sum graph of R and vice versa.