THE TOTAL TORSION ELEMENT GRAPH WITHOUT THE ZERO ELEMENT OF MODULES OVER COMMUTATIVE RINGS
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Let M be a module over a commutative ring R, and let T(M) be its set of torsion elements. The total torsion element graph of M over R is the graph T(Γ(M)) with vertices all elements of M, and two distinct vertices m and n are adjacent if and only if m + n ∈ T(M). In this paper, we study the basic properties and possible structures of two (induced) subgraphs Tσr0 (Γ(M)) and T0 (Γ(M)) of T(Γ(M)), with vertices T(M) ＼ {0} and M ＼ {0}, respectively. The main purpose of this paper is to extend the definitions and some results given in [6] to a more general total torsion element graph case.
Abstract
Let M be a module over a commutative ring R, and let T(M) be its set of torsion elements. The total torsion element graph of M over R is the graph T(Γ(M)) with vertices all elements of M, and two distinct vertices m and n are adjacent if and only if m + n ∈ T(M). In this paper, we study the basic properties and possible structures of two (induced) subgraphs Tσr0 (Γ(M)) and T0 (Γ(M)) of T(Γ(M)), with vertices T(M) ＼ {0} and M ＼ {0}, respectively. The main purpose of this paper is to extend the definitions and some results given in [6] to a more general total torsion element graph case.
참고문헌 (15)
- Ahmad Abbasi, "The total graph of a commutative ring with respect to proper ideals", 대한수학회지, 49(1), pp.85~98 (2012), 대한수학회
- A. Abbasi, "The total graph of a module over a commutative ring with respect to proper submodules", J. Algebra Appl, 11(3), pp.13~ (2012)
- D. D. Anderson, "The set of torsion elements of a module", Comm. Algebra, 42, pp.1835~1843 (2014)
- D. D. Anderson, "Beck's coloring of a commutative ring", J. Algebra, 159(2), pp.500~514 (1993)
- D. F. Anderson, "Zero-divisor graphs in commutative rings, in Commutative Algebra, Noetherian and Non-Noetherian Perspectives", pp.23~45 (2011), Springer-Verlag
- D. F. Anderson, "On the total graph of a commutative ring without the zero element", J. Algebra Appl, 11(4), pp.18~ (2012)
- D. F. Anderson, "The generalized total graph of a commutative ring", J. Algebra Appl, 12(5), pp.18~ (2013)
- D. F. Anderson, "The total graph of a commutative ring", J. Algebra, 320(7), pp.2706~2719 (2008)
- D. F. Anderson, "zero-divisor graph of a commutative ring", J. Algebra, 217(2), pp.437~447 (1999)
- D. F. Anderson, "On the diameter and girth of a zero-divisor graph", J. Pure Appl. Algebra, 210(2), pp.543~550 (2007)
- Z. Barati, "On the associated graphs to a commutative ring", J. Algebra Appl, 11(2), pp.17~ (2012)
- I. Beck, "Coloring of commutative rings", J. Algebra, 116(1), pp.208~226 (1988)
- S. Ebrahimi Atani, "The total torsion element graph of semimodules over commutative semirings", Algebra and Discrete Mathematics, 16(1), pp.1~15 (2013)
- S. Ebrahimi Atani, "The total torsion element graph of a module over a commutative ring", An. St. Univ. Ovidius Constant, 19(1), pp.23~34 (2011)
- R. L. MacCasland, "Prime submodules of Noetherian modules", Rocky Mountain J. Math, 23, pp.1041~1062 (1993)