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On the Annihilator Submodules and the Annihilator Essential Graph

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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).

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Acknowledgements

We would like to thank the referee for a careful reading of our article and insightful comments which saved us from several errors.

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Authors and Affiliations

  1. Department of Mathematics, Imam Khomeini International University, 34149-1-6818, Qazvin, Iran

    Sakineh Babaei & Shiroyeh Payrovi

  2. Eski Silahtaraga Elektrik Santrali, Kazim Karabekir, Istanbul Bilgi University, Cad. No: 2/1334060, Eyup, Istanbul, Turkey

    Esra Sengelen Sevim

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  1. Sakineh Babaei
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  2. Shiroyeh Payrovi
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  3. Esra Sengelen Sevim
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Correspondence to Shiroyeh Payrovi.

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Babaei, S., Payrovi, S. & Sevim, E.S. On the Annihilator Submodules and the Annihilator Essential Graph. Acta Math Vietnam 44, 905–914 (2019). https://doi.org/10.1007/s40306-018-00306-1

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  • DOI: https://doi.org/10.1007/s40306-018-00306-1

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