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


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).


Annihilator submodule Annihilator essential graph Zero divisor graph 

Mathematics Subject Classification (2010)

13A15 05C99 



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


  1. 1.
    El-Bast, Z.A., Smith, P.F.: Multiplication modules. Comm. Algebra 16(4), 755–779 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anderson, D.F., LaGrange, J.D.: Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph. J. Pure Appl. Algebra 216(7), 1626–1636 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217(2), 434–447 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anderson, D.D., Chun, S.: The set of torsion elements of a module. Comm. Algebra 42(4), 1835–1843 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Badawi, A.: On the annihilator graph of a commutative ring. Comm. Algebra 42(1), 108–121 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beck, I.: Coloring of commutative rings. J. Algebra 116(1), 208–226 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Coykendall, J., Sather-Wagstaff, S., Sheppardson, L., Spiroff, S.: On Zero Divisor Graphs Progress in Commutative Algebra, vol. II. Walter de Gruyter, Berlin (2012)zbMATHGoogle Scholar
  8. 8.
    Lu, C.-P.: Unions of prime submodules. Houston J. Math. 23(2), 203–213 (1997)MathSciNetzbMATHGoogle Scholar
  9. 9.
    McCasland, R.L., Moore, M.E., Smith, P.F.: On the spectrum of a module over a commutative ring. Comm. Algebra 25(1), 79–103 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mulay, S.B.: Cycles and symmetries of zero-divisors. Comm. Algebra 30(7), 3533–3558 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Spiroff, S., Wickham, C.: A zero divisor graph determine by equivalence classes of zero divisors. Comm. Algebra 39(7), 2338–2348 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

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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