The connected subgraph of the torsion graph of a module
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper, we will investigate the concept of the torsion-graph of an R-module M, in which the set T(M)*makes up the vertices of the corresponding torsion graph, Γ(M), with any two distinct vertices forming an edge if [x : M][y : M]M = 0. We prove that, if Γ(M) contains a cycle, then gr(Γ(M)) 4 and Γ(M) has a connected induced subgraph Γ(M) with vertex set fm 2 T(M) j Ann(m)M ̸= 0g and diam(Γ(M)) ≤3. Moreover, if M is a multiplication R-module, then Γ(M) is a maximal connected subgraph of Γ(M). Also Γ(M) and Γ(S-1M) are isomorphic graphs, where S = R n Z(M). Furthermore, we show that, if Γ(M) is uniquely complemented, then S-1M is a von Neumann regular module or Γ(M) is a star graph.
Abstract
In this paper, we will investigate the concept of the torsion-graph of an R-module M, in which the set T(M)*makes up the vertices of the corresponding torsion graph, Γ(M), with any two distinct vertices forming an edge if [x : M][y : M]M = 0. We prove that, if Γ(M) contains a cycle, then gr(Γ(M)) 4 and Γ(M) has a connected induced subgraph Γ(M) with vertex set fm 2 T(M) j Ann(m)M ̸= 0g and diam(Γ(M)) ≤3. Moreover, if M is a multiplication R-module, then Γ(M) is a maximal connected subgraph of Γ(M). Also Γ(M) and Γ(S-1M) are isomorphic graphs, where S = R n Z(M). Furthermore, we show that, if Γ(M) is uniquely complemented, then S-1M is a von Neumann regular module or Γ(M) is a star graph.
참고문헌 (22)
- D. D. Anderson, "Multiplication ideals, multiplication rings and the ring R(X)", Canad. J. Math, 28(4), pp.260~768 (1976)
- D. D. Anderson, "Beck's coloring of a commutative rings", J. Algebra, 159(2), pp.500~514 (1993)
- D. F. Anderson, "Zero-divisor graphs in commuta- tive rings, Commutative algebra, Noetherian and non-Noetherian perspectives", pp.23~45 (2011), Springer
- D. F. Anderson, "The total graph of a commutative ring", J. Algebra, 320(7), pp.2706~2719 (2008)
- D. F. Anderson, "On the zero-divisor graph of a ring", Comm. Algebra, 36(8), pp.3073~3092 (2008)
- D. F. Anderson, "The zero-divisor graph of a commutative ring. II, Ideal theoretic methods in commutative algebra", Lecture Notes in Pure and Appl. Math, 220, pp.61~72 (2001), Dekker
- D. F. Anderson, "Zero-divisor graphs, von Neumann regular rings, and Boolean algebras", J. Pure Appl. Algebra, 180(3), pp.221~241 (2003)
- D. F. Anderson, "The zero-divisor graph of a commutative ring", J. Algebra, 217(2), pp.434~447 (1999)
- A. Badawi, "Divisibility conditions in commutative rings with zero- divisors", Comm. Algebra, 38(8), pp.4031~4047 (2002)
- A. Barnard, "Multiplication modules", J. Algebra, 71(1), pp.174~178 (1981)
- I. Beck, "Coloring of commutative rings", J. Algebra, 116(1), pp.208~226 (1988)
- G. A. Cannon, "Zero-divisor graphs of nearrings and semigroups, Nearrings and nearfields", pp.189~200 (2005), Springer
- F. R. DeMeyer, "The zero-divisor graph of a commutative semigroup", Semigroup Forum, 65(2), pp.206~214 (2002)
- R. Diestel, "Graph Theory" (1997), Springer-Verlag
- Z. A. El-Bast, "Multiplication modules", Comm. Algebra, 16(4), pp.755~779 (1988)
- SH. Ghalandarzadeh, "Torsion graph over multiplication modules", Extracta Math, 24(3), pp.281~299 (2009)
- 한준철, "The zero-divisor graph under group actions in a noncommutative ring", 대한수학회지, 45(6), pp.1647~1659 (2008), 대한수학회
- F. Kash, "Modules and Rings" (1982), Academic Press
- P. Malakooti Rad, "On the torsion graph and von Neumann regular rings", Filomat, 26, pp.47~53
- H. Matsumara, "Commutative Ring Theory" (1986), Cambridge University Press
- S. P. Redmond, "The zero-divisor graph of a noncommutative ring", Internat. J. Commutative Rings, 1(4), pp.203~211 (2002)
- P. Ribenboim, "Algebraic Numbers" (1972), Wiley