Some results on a graph associated with a non-quasi-local atomic domain

Abstract

Let R be an atomic domain which admits at least two maximal ideals. Let Irr(R) denote the set of all irreducible elements of R and let A(R) = {Rπ | π ∈ Irr(R)}. Let I(R) denote the subset of A(R) consisting of all Rπ ∈ A(R) such that π does not belong to the Jacobson radical of R. With R, we associate an undirected graph denoted by G(R) whose vertex set is I(R) and distinct vertices Rπ₁ and Rπ₂ are adjacent if and only if Rπ₁ ∩ Rπ₂ = Rπ₁π₂. The aim of this article is to discuss some results on the connectedness of G(R) and on the girth of G(R).