On the Annihilator Submodules and the Annihilator Essential Graph
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On the Annihilator Submodules and the Annihilator Essential Graph Sakineh Babaei1 · Shiroyeh Payrovi1 · Esra Sengelen Sevim2 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, AnnM (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, AER (M), which is constructed from classes of zero divisors, determined by annihilator submodules of M and distinct vertices [a] and [b] are adjacent whenever AnnM (a) + AnnM (b) is an essential submodule of M. Among other things, we determine when AER (M) is a connected graph, a star graph, or a complete graph. We compare the clique number of AER (M) and the cardinal of m − AssR (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 ∈ AnnR (M/P ) = {r ∈ R : rM ⊆ P }. Let SpecR (M) denote the set of prime submodules of M. For a ∈ R Shiroyeh Payrovi
[email protected] Sakineh Babaei [email protected] Esra Sengelen Sevim [email protected] 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
S. Babaei et al.
we call AnnM (a) = {m ∈ M : am = 0} the annihilator submodule of a in M. Let m − AssR (M) = {P ∈ SpecR (M) : P = AnnM (a), for some 0 = a ∈ R}. The properties of prime submodules and m − AssR (M) are studied in [8, 9] and [4]. By [8, Proposition 3.2], any maximal element of {AnnM (a) : a ∈ AnnR (M)} is a prime submodule of M. Thus, m − AssR (M) is a non-empty set, when M is a Noetherian R-module. In Section 2, we study some properties of the elements of m − AssR (M). In particular, we show that AnnM (a) = {m ∈ M | rm ∈ AnnR (aM)M for some r ∈ AnnR (aM)} whenever AnnM (a) is a prime submodule of M and a ∈ r(AnnR (M)). Also, we compare m − AssR (M) and the set of associated prime ideals of R, AssR (R), and we show that: m − AssR (M) = {AnnM (a) | AnnR (a) ∈ AssR (R)} , where M is either a free or a faithful multiplication R-module. There are many studies of various graphs associated to rings or modules (see for instance [3, 5, 6, 10]). A submodule N of M is called an essential submodule if it has a non-zero intersection with any other non-zero submodule of M. In the third section, we investigate the property of being essential for an annihilator submodule, AnnM (a), in two cases, a ∈ r(AnnR (M)) = {r ∈ R : r t M = 0 for some posit
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