The principal small intersection graph of a commutative ring

Let R be a commutative ring with non-zero identity.The small intersection graph of R, denoted by G(R), is a graph with the vertex set V (G(R)), where V (G(R)) is the set of all proper non-small ideals of R and two distinct vertices I and J are adjacent if and only if I ∩ J is not small in R.In this paper, we introduce a certain subgraph P G(R) of G(R), called the principal small intersection graph of R. It is the subgraph of G(R) induced by the set of all proper principal non-small ideals of R. We study the diameter, the girth, the clique number, the independence number and the domination number of P G(R).Moreover, we present some results on the complement of the principal small intersection graph.