The M-intersection graph of ideals of a commutative ring.

Let R be a commutative ring and M be an R-module, and let I(R)∗ be the set of all nontrivial ideals of R. The M-intersection graph of ideals of R, denoted by GM(R), is a graph with the vertex set I(R)∗, and two distinct vertices I and J are adjacent if and only if IM∩JM≠{0}. For every multiplication R-module M, the diameter and the girth of GM(R) are determined. Among other results, we prove that if M is a faithful R-module and the clique number of GM(R) is finite, then R is a semilocal ring. We denote the ℤn-intersection graph of ideals of the ring ℤm by Gn(ℤm), where n,m≥2 are integers and ℤn is a ℤm-module. We determine the values of n and m for which Gn(ℤm) is perfect. Furthermore, we derive a sufficient condition for Gn(ℤm) to be weakly perfect. [ABSTRACT FROM AUTHOR]