Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ZERO DIVISOR GRAPHS OF SKEW GENERALIZED POWER SERIES RINGS

  • MOUSSAVI, AHMAD (Department of Pure Mathematics Faculty of Mathematical Sciences Tarbiat Modares University) ;
  • PAYKAN, KAMAL (Department of Pure Mathematics Faculty of Mathematical Sciences Tarbiat Modares University)
  • Received : 2015.03.13
  • Published : 2015.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let R be a ring, (S,${\leq}$) a strictly ordered monoid and ${\omega}$ : S ${\rightarrow}$ End(R) a monoid homomorphism. The skew generalized power series ring R[[S,${\omega}$]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we investigate the interplay between the ring-theoretical properties of R[[S,${\omega}$]] and the graph-theoretical properties of its zero-divisor graph ${\Gamma}$(R[[S,${\omega}$]]). Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.

Keywords

References

  1. M. Afkhami, K. Khashyarmanesh, and M. R. Khorsandi, Zero-divisor graphs of Ore extension rings, J. Algebra Appl. 10 (2011), no. 6, 1309-1317. https://doi.org/10.1142/S0219498811005191
  2. S. Akbari and A. Mohammadian, Zero-divisor graphs of non-commutative rings, J. Algebra 296 (2006), no. 2, 462-479. https://doi.org/10.1016/j.jalgebra.2005.07.007
  3. D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 (1993), no. 2, 500-514. https://doi.org/10.1006/jabr.1993.1171
  4. 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
  5. D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210 (2007), no. 2, 543-550. https://doi.org/10.1016/j.jpaa.2006.10.007
  6. M. Axtell, J. Coykendall, and J. Stickles, Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra 33 (2005), no. 6, 2043-2050. https://doi.org/10.1081/AGB-200063357
  7. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
  8. G. M. Bergman and I. M. Issacs, Rings with fixed-point-free group actions, Proc. London Math. Soc. 27 (1973), 69-87.
  9. G. F. Birkenmeier and J. K. Park, Triangular matrix representations of ring extensions, J. Algebra 265 (2003), no. 2, 457-477. https://doi.org/10.1016/S0021-8693(03)00155-8
  10. P. M. Cohn, A Morita context related to finite automorphism groups of rings, Pacific J. Math. 98 (1982), no. 1, 37-54. https://doi.org/10.2140/pjm.1982.98.37
  11. P. M. Cohn, Free Rings and Their Relations, Second Ed., Academic Press, London, 1985.
  12. P. M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), no. 6, 641-648. https://doi.org/10.1112/S0024609399006116
  13. G. A. Elliott and P. Ribenboim, Fields of generalized power series, Arch. Math. 54 (1990), no. 4, 365-371. https://doi.org/10.1007/BF01189583
  14. S. P. Farbman, The unique product property of groups and their amalgamated free products, J. Algebra 178 (1995), no. 3, 962-990. https://doi.org/10.1006/jabr.1995.1385
  15. J. W. Fisher and S. Montgomery, Semiprime skew group rings, J. Algebra 52 (1978), no. 1, 241-247. https://doi.org/10.1016/0021-8693(78)90272-7
  16. E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 107 (2005), no. 3, 207-224. https://doi.org/10.1007/s10474-005-0191-1
  17. I. Kaplansky, Commutative Rings, Rev. Ed. Chicago: Univ. of Chicago Press, 1974.
  18. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300.
  19. T. Y. Lam, A First Course in Noncommutative Rings, New York, Springer-Verlag, 1991.
  20. Z. K. Liu, Endomorphism rings of modules of generalized inverse polynomials, Comm. Algebra 28 (2000), no. 2, 803-814. https://doi.org/10.1080/00927870008826861
  21. Z. K. Liu, Triangular matrix representations of rings of generalized power series, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 4, 989-998. https://doi.org/10.1007/s10114-005-0555-z
  22. T. Lucas, The diameter of a zero divisor graph, J. Algebra 301 (2006), no. 1, 174-193. https://doi.org/10.1016/j.jalgebra.2006.01.019
  23. G. Marks, R. Mazurek, and M. Ziembowski, A new class of unique product monoids with applications to ring theory, Semigroup Forum 78 (2009), no. 2, 210-225. https://doi.org/10.1007/s00233-008-9063-7
  24. G. Marks, R. Mazurek, and M. Ziembowski, A unified approach to various generalizations of Armendariz rings, Bull. Aust. Math. Soc. 81 (2010), no. 3, 361-397. https://doi.org/10.1017/S0004972709001178
  25. R. Mazurek and M. Ziembowski, On von Neumann regular rings of skew generalized power series, Comm. Algebra 36 (2008), no. 5, 1855-1868. https://doi.org/10.1080/00927870801941150
  26. S. Montgomery, Outer automorphisms of semi-prime rings, J. London Math. Soc. 18 (1978), no. 2, 209-220.
  27. P. P. Nielsen, Semi-commutativity and the McCoy condition, J. Algebra 298 (2006), no. 1, 134-141. https://doi.org/10.1016/j.jalgebra.2005.10.008
  28. J. Okninski, Semigroup Algebras, Marcel, Dekker, 1991.
  29. D. S. Passman, The Algebraic Structure of Group Rings, Wiley, 1977.
  30. K. Paykan and A. Moussavi, McCoy property and nilpotent elements of skew generalized power series rings, preprint.
  31. K. Paykan, A. Moussavi, and M.Zahiri, Special Properties of rings of skew generalized power series, Comm. Algebra 42 (2014), no. 12, 5224-5248 https://doi.org/10.1080/00927872.2013.836532
  32. S. P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings 1 (2002), 203-211.
  33. S. P. Redmond, Structure in the zero-divisor graph of a non-commutative ring, Houston J. Math. 30 (2004), no. 2, 345-355.
  34. P. Ribenboim, Rings of generalized power series II: Units and Zero-Divisors, J. Algebra 168 (1994), no. 1, 71-89. https://doi.org/10.1006/jabr.1994.1221
  35. P. Ribenboim, Special properties of generalized power series, J. Algebra 173 (1995), no. 3, 566-586. https://doi.org/10.1006/jabr.1995.1103
  36. P. Ribenboim, Some examples of valued fields, J. Algebra 173 (1995), no. 3, 668-678. https://doi.org/10.1006/jabr.1995.1108
  37. P. Ribenboim, Semisimple rings and von Neumann regular rings of generalized power series, J. Algebra 198 (1997), no. 2, 327-338. https://doi.org/10.1006/jabr.1997.7063
  38. A. A. Tuganbaev, Some ring and module properties of skew Laurent series, Formal power series and algebraic combinatorics, (Moscow), 613-622, Springer, Berlin, 2000.
  39. S. E. Wright, Lengths of paths and cycles in zero-divisor graphs and digraphs of semigroups, Comm. Algebra 35 (2007), no. 6, 1987-1991. https://doi.org/10.1080/00927870701247146