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On the Annihilator Submodules and the Annihilator Essential Graph
Sakineh Babaei
,
Shiroyeh Payrovi
,
Esra Sengelen Sevim
Abstract
Let $R$ be a commutative ring and let $M$ be an $R$-module. For $a \in R$, $\mathrm{Ann}_M(a) = \{m \in M \,:\, am = 0\}$ 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 $M$, $\mathrm{AE}_R(M)$, which is constructed from classes of zero divisors, determined by annihilator submodules of $M$ and distinct vertices $[a]$ and $[b]$ are adjacent whenever $\mathrm{Ann}_M(a) + \mathrm{Ann}M(b)$ is an essential submodule of $M$. Among other things, we determine when $\mathrm{AE}_R(M)$ is a connected graph, a star graph, or a complete graph. We compare the clique number of $\mathrm{AE}_R(M)$ and the cardinal of $-\mathrm{Ass}_R(M)$.