Some results on a subgraph of the intersection graph of ideals of a commutative ring

The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal.   Let R be a ring. Let us denote  the collection  of all proper ideals  of R by \mathbb{I}(R)  and \mathbb{I}(R)\backslash \{(۰)\} by \mathbb{I}(R)^{*}.  With R, we associate an undirected graph denoted by g(R), whose vertex set is \mathbb{I}(R)^{*} and distinct vertices I_{۱}, I_{۲} are adjacent in g(R)  if and only if I_{۱}\cap I_{۲}\neq I_{۱}I_{۲}.  The aim of this article is to study the interplay between the graph-theoretic properties of g(R) and the ring-theoretic properties of R.

Keywords:

Artinian ring , Special principal ideal ring , diameter , girth , clique number