Extended Idempotent Divisor Graph of Commutative Rings

Abstract

Associate graph Л(R) is said to be idempotent divisor graph with vertices set V(Л(R))=R^*, if any two non- zero elements a_1 and a_2  are adjacent if and only if a_1.a_2=e, where e is an idempotent element not equal 1. In this work we study and introduce the extended idempotent divisor graph that is for any  two non-zero elements  a_1 and a_2  adjacent if 〖〖 a〗_1〗^(t_1 ). 〖a_2〗^(t_2 )=e , where t_1,t_2  ∈Z and e an idempotent element not equal one, and we give some results for properties such as diameter and the girth of this graph. Also, we investigated rings isomorphic to direct product two finite local rings.