ON A GRAPH OF IDEALS OF A COMMUTATIVE RING

Abstract

In this paper, we introduce and investigate a new graph of a commutative ring R, denoted by G(R), with all nontrivial ideals of R as vertices,and two distinct vertices I and J are adjacent if and only if ann(I J) =ann(I) + ann(J). In this article, the basic properties and possible structuresof the graph G(R) are studied and investigated as diameter, girth, clique number, cut vertex and domination number. We characterize all rings R for whichG(R) is planar, complete and complete r-partite. We show that, if (R; M) isa local Artinian ring, then G(R) is complete if and only if Soc(R) is simple.Also, it is shown that if R is a ring with G(R) is r-regular, then either G(R) iscomplete or null graph. Moreover, we show that if R is an Artinian ring, thenR is a serial ring if and only if G(R=I) is complete for each ideal I of R.