Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A TORSION GRAPH DETERMINED BY EQUIVALENCE CLASSES OF TORSION ELEMENTS AND ASSOCIATED PRIME IDEALS

  • Reza Nekooei (Department of Pure Mathematics Mahani Mathematical Research Center Shahid Bahonar University of Kerman) ;
  • Zahra Pourshafiey (Department of Pure Mathematics Mahani Mathematical Research Center Shahid Bahonar University of Kerman)
  • Received : 2023.07.12
  • Accepted : 2023.11.03
  • Published : 2024.05.31
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Abstract

In this paper, we define the torsion graph determined by equivalence classes of torsion elements and denote it by AE(M). The vertex set of AE(M) is the set of equivalence classes {[x] | x ∈ T(M)*}, where two torsion elements x, y ∈ T(M)* are equivalent if ann(x) = ann(y). Also, two distinct classes [x] and [y] are adjacent in AE(M), provided that ann(x)ann(y)M = 0. We shall prove that for every torsion finitely generated module M over a Dedekind domain R, a vertex of AE(M) has degree two if and only if it is an associated prime of M.

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References

  1. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434-447. https://doi.org/10.1006/jabr.1998.7840 
  2. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, MA, 1969. 
  3. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226. https://doi.org/10.1016/0021-8693(88)90202-5 
  4. M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011), no. 4, 727-739. https://doi.org/10.1142/S0219498811004896 
  5. M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2011), no. 4, 741-753. https://doi.org/10.1142/S0219498811004902 
  6. S. Ghalandarzadeh and P. Malakooti Rad, Torsion graph over multiplication modules, Extracta Math. 24 (2009), no. 3, 281-299. 
  7. N. Jacobson, Basic Algebra. II, Second Edition, New York, 2009. 
  8. S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra 30 (2002), no. 7, 3533-3558. https://doi.org/10.1081/AGB-120004502 
  9. S. Spiroff and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra 39 (2011), no. 7, 2338-2348. https://doi.org/10.1080/00927872.2010.488675