Prime ideal sum graph of a commutative ring

Abstract

Let R be a commutative ring with identity. The prime ideal sum graph of R, denoted by PIS(R), is a graph whose vertices are nonzero proper ideals of R and two distinct vertices I and J are adjacent if and only if I J is a prime ideal of R. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. The clique number, the chromatic number and the domination number of the prime ideal sum graph for some classes of rings are studied. It is observed that under which condition PIS(R) is complete. Moreover, the diameter and the girth of PIS(R) are studied.