The complement of the 𝑀-intersection graph of ideals of a ring

Let 𝑅 be a commutative ring with identity and 𝑀 be a unitary 𝑅-module,and let 𝐼(𝑅)* be the set of all nontrivial ideals of 𝑅. The complement ofthe 𝑀-intersection graph of ideals of 𝑅, denoted by Γ𝑀(𝑅), is a graphwith the vertex set 𝐼(𝑅)* , and two distinct vertices 𝐼 and 𝐽 are adjacentif and only if 𝐼𝑀∩𝐽𝑀={0}. In this paper, for every multiplication𝑅-module 𝑀, the diameter and the girth of Γ𝑀(𝑅) are determined. Also,we show that if 𝑚,𝑛>1 are two integers and ℤ𝑛 is a ℤ𝑚-module, thenthe complement of the ℤ𝑛-intersection graph of ideals of ℤ𝑚 is weakly perfect.