On the Annihilator Submodules and the Annihilator Essential Graph

Let R be a commutative ring and let M be an R-module. For a is an element of R, Ann(M)(a) = {m is an element of M : am = 0} is said to be an annihilator submodule of M. In this paper, we study the property of being prime or essential for annihilator submodules of M. Also, we introduce the annihilator essential graph of equivalence classes of zero divisors of M, AE(R)(M), which is constructed from classes of zero divisors, determined by annihilator submodules of M and distinct vertices [a] and [b] are adjacent whenever Ann(M)(a) + Ann(M)(b) is an essential submodule of M. Among other things, we determine when AE(R)(M) is a connected graph, a star graph, or a complete graph. We compare the clique number of AE(R)(M) and the cardinal of m -Ass(R)(M).