A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS

Abstract

Let R be a commutative ring with nonzero identity and let M be a unitary R-module. The essential graph of M , denoted by EG(M) is a simple undirected graph whose vertex set is Z(M)\AnnR(M) and two distinct vertices x and y are adjacent if and only if AnnM (xy) is an essential submodule of M . Let r(AnnR(M)) 6= AnnR(M). It is shown that EG(M) is a connected graph with diam(EG(M)) ≤ 2. Whenever M is Noetherian, it is shown that EG(M) is a complete graph if and only if either Z(M) = r(AnnR(M)) or EG(M) = K2 and diam(EG(M)) = 2 if and only if there are x, y ∈ Z(M)\AnnR(M) and p ∈ AssR(M) such that xy 6∈ p. Moreover, it is proved that gr(EG(M)) ∈ {3,∞}. Furthermore, for a Noetherian module M with r(AnnR(M)) = AnnR(M) it is proved that |AssR(M)| = 2 if and only if EG(M) is a complete bipartite graph that is not a star. Mathematics Subject Classification (2020): 05C25, 13C99