Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON SOME PROPERTIES OF MALCEV-NEUMANN MODULES

  • Zhao, Renyu (COLLEGE OF ECONOMICS AND MANAGEMENT NORTHWEST NORMAL UNIVERSITY) ;
  • Liu, Zhongkui (COLLEGE OF ECONOMICS AND MANAGEMENT NORTHWEST NORMAL UNIVERSITY)
  • Published : 2008.08.31
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Abstract

Let M be a right R-module, G an ordered group and ${\sigma}$ a map from G into the group of automorphisms of R. The conditions under which the Malcev-Neumann module M* ((G)) is a PS module and a p.q.Baer module are investigated in this paper. It is shown that: (1) If $M_R$ is a reduced ${\sigma}$-compatible module, then the Malcev-Neumann module M* ((G)) over a PS-module is also a PS-module; (2) If $M_R$ is a faithful ${\sigma}$-compatible module, then the Malcev-Neumann module M* ((G)) is a p.q.Baer module if and only if the right annihilator of any G-indexed family of cyclic submodules of M in R is generated by an idempotent of R.

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References

  1. G. F. Birkenmeier, J. Y. Kim, and J. K. Park , On polynomial extensions of principally quasi-Baer rings, Kyungpook Math. J. 40 (2000), no. 2, 247-253
  2. G. F. Birkenmeier, J. Y. Kim, and J. K. Park, Principally quasi-Baer rings, Comm. Algebra 29 (2001), no. 2, 639-660 https://doi.org/10.1081/AGB-100001530
  3. G. F. Birkenmeier, J. Y. Kim, and J. K. Park, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra 159 (2001), no. 1, 25-42 https://doi.org/10.1016/S0022-4049(00)00055-4
  4. W. E. Clark, Twisted matrix units semigroup algebras, Duck Math. J. 34 (1967), 417-423
  5. I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968
  6. T. K. Lee and Y. Q. Zhou, Reduced modules, Rings, Modules, Algebras and Abelian Groups, 365-377. Lecture Notes in Pure and Appl. Math. 236, Dekker, New York, 2004
  7. Z. K. Liu, PS-modules over rings of generalized power series, Northeast. Math. J. 18 (2002), no. 3, 254-260
  8. Z. K. Liu, A note on principally quasi-Baer rings, Comm. Algebra 30 (2002), no. 8, 3885-3890 https://doi.org/10.1081/AGB-120005825
  9. M. Lorenz, Division algebras generated by finitely generated nilpotent groups, J. Algebra 85 (1983), no. 2, 368-381 https://doi.org/10.1016/0021-8693(83)90101-1
  10. L. Makar-Limanov, The skew field of fractions of the Weyl algebra contains a free noncommutative subalgebra, Comm. Algebra 11 (1983), no. 17, 2003-2006 https://doi.org/10.1080/00927878308822945
  11. I. Musson and K. Stafford, Malcev-Neumann group rings, Comm. Algebra 21 (1993), no. 6, 2065-2075 https://doi.org/10.1080/00927879308824665
  12. W. K. Nicholson and J. F. Watters, Rings with projective socle, Proc. Amer. Math. Soc. 102 (1988), no. 3, 443-450
  13. D. Passman, The Algebraic Structure of Group Rings, Pure and Applied Mathematics. Wiley-Interscience John Wiley & Sons, New York-London-Sydney, 1977
  14. C. Sonin, Krull dimension of Malcev-Neumann rings, Comm. Algebra 26 (1998), no. 9, 2915-2931 https://doi.org/10.1080/00927879808826317
  15. W. M. Xue, Modules with projective socles, Riv. Mat. Univ. Parma (5) 1 (1992), 311-315

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