Simple-intersection graphs of rings

Abstract

<abstract><p>Let $ R $ be a ring with unity. In this paper, we introduce a new graph associated with $ R $ called the simple-intersection graph of $ R $, denoted by $ GS(R) $. The vertices of $ GS(R) $ are the nonzero ideals of $ R $, and two vertices are adjacent if and only if their intersection is a nonzero simple ideal. We study the interplay between the algebraic properties of $ R $, and the graph properties of $ GS(R) $ such as connectedness, bipartiteness, girth, dominating sets, etc. Moreover, we determine the precise values of the girth and diameter of $ GS(R) $, as well as give a formula to compute the clique and domination numbers of $ GS(R) $.</p></abstract>