Acta Mathematica Vietnamica https://doi.org/10.1007/s40306-018-00306-1 On the Annihilator Submodules and the Annihilator Essential Graph Sakineh Babaei 1 · Shiroyeh Payrovi 1 · Esra Sengelen Sevim 2 Received: 20 November 2017 / Revised: 12 May 2018 / Accepted: 28 June 2018 / © Institute of Mathematics. 2019 Abstract Let R be a commutative ring and let M be an R-module. For a ∈ R, Ann M (a) ={m ∈ 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). Keywords Annihilator submodule · Annihilator essential graph · Zero divisor graph Mathematics Subject Classification (2010) 13A15 · 05C99 1 Introduction Throughout this paper, R is a commutative ring with non-zero identity and all modules are unitary. Let M be an R-module. A proper submodule P of M is said to be prime if rm ∈ P for r ∈ R and m ∈ M, implies that m ∈ P or r ∈ Ann R (M/P) ={r ∈ R : rM ⊆ P }. Let Spec R (M) denote the set of prime submodules of M. For a ∈ R  Shiroyeh Payrovi shpayrovi@sci.ikiu.ac.ir Sakineh Babaei sakinehbabaei@gmail.com Esra Sengelen Sevim esra.sengelen@bilgi.edu.tr 1 Department of Mathematics, Imam Khomeini International University, 34149-1-6818, Qazvin, Iran 2 Eski Silahtaraga Elektrik Santrali, Kazim Karabekir, Istanbul Bilgi University, Cad. No: 2/1334060, Eyup, Istanbul, Turkey