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Acta Mathematica Vietnamica

, Volume 44, Issue 4, pp 905–914 | Cite as

On the Annihilator Submodules and the Annihilator Essential Graph

  • Sakineh Babaei
  • Shiroyeh PayroviEmail author
  • Esra Sengelen Sevim
Article
  • 51 Downloads

Abstract

Let R be a commutative ring and let M be an R-module. For aR, AnnM(a) = {mM : 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 

Notes

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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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Sakineh Babaei
    • 1
  • Shiroyeh Payrovi
    • 1
    Email author
  • Esra Sengelen Sevim
    • 2
  1. 1.Department of MathematicsImam Khomeini International UniversityQazvinIran
  2. 2.Eski Silahtaraga Elektrik Santrali, Kazim KarabekirIstanbul Bilgi UniversityEyupTurkey

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