On the intersection graph of ideals of commutative rings

Abstract

Let R be a ring and I^∗(R) be the set of all non-trivial ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I^∗(R) and two distinct vertices I and J are adjacent if and only if I ∪ J ≠ 0. In this paper, it is shown that if (R, m) is a Noetherian local ring and m is a principal ideal, then G(R) is complete if and only if either R is an integral domain or Artinian. The intersection of all non-zero ideals of a ring R is called the heart of R. Among other results, we prove that every Noetherian ring with non-zero heart is Artinian.