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Baer Group Rings with Involution

Year 2017, Volume: 22 Issue: 22, 1 - 10, 11.07.2017
https://doi.org/10.24330/ieja.325913

Abstract

We prove that if a group ring $RG$ is a (quasi) Baer $*$-ring, then so is $R$, whereas converse is not true.
      Sufficient conditions are given so that for some finite cyclic groups $G$,
     if $R$ is (quasi-) Baer $*$-ring, then so is the group ring $RG$.
     We prove that if the group ring $RG$ is a Baer $*$-ring, then so is $RH$ for every subgroup $H$ of $G$.
     Also, we generalize results of Zhong Yi, Yiqiang Zhou (for (quasi-) Baer rings) and  L. Zan, J. Chen
      (for principally quasi-Baer and principally projective rings).

References

  • E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc., 18 (1974), 470-473.
  • G. F. Birkenmeier, J. Y. Kim and J. K. Park, On polynomial extensions of principally quasi-Baer rings, Kyungpook Math. J., 40(2) (2000), 247-253.
  • G. F. Birkenmeier, J. Y. Kim and J. K. Park, Principally quasi-Baer rings, Comm. Algebra, 29(2) (2001), 639-660.
  • G. F. Birkenmeier, J. Y. Kim and J. K. Park, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra, 159(1) (2001), 25-42.
  • G. F. Birkenmeier and J. K. Park, Triangular matrix representations of ring extensions, J. Algebra, 265(2) (2003), 457-477.
  • W. E. Clark, Twisted matrix units semigroup algebras, Duke Math. J., 34(3) (1967), 417-423.
  • N. J. Groenewald, A note on extensions of Baer and p.p.-rings, Publ. Inst. Math. (Beograd) (N.S.), 34(48) (1983), 71-72.
  • Y. Hirano, On ordered monoid rings over a quasi-Baer ring, Comm. Algebra, 29(5) (2001), 2089-2095.
  • I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York- Amsterdam, 1968.
  • A. Khairnar and B. N. Waphare, Order properties of generalized projections, Linear Multilinear Algebra, 65(7) (2017), 1446-1461.
  • B. N. Waphare and A. Khairnar, Semi-Baer modules, J. Algebra Appl., 14(10) (2015), 1550145 (12 pp).
  • Z. Yi and Y. Zhou, Baer and quasi-Baer properties of group rings, J. Aust. Math. Soc., 83(2) (2007), 285-296.
  • L. Zan and J. Chen, p.p. properties of group rings, Int. Electron. J. Algebra, 3 (2008), 117-124.
  • L. Zan and J. Chen, Principally quasi-Baer properties of group rings, Studia Sci. Math. Hungar., 49(4) (2012), 454-465.
Year 2017, Volume: 22 Issue: 22, 1 - 10, 11.07.2017
https://doi.org/10.24330/ieja.325913

Abstract

References

  • E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc., 18 (1974), 470-473.
  • G. F. Birkenmeier, J. Y. Kim and J. K. Park, On polynomial extensions of principally quasi-Baer rings, Kyungpook Math. J., 40(2) (2000), 247-253.
  • G. F. Birkenmeier, J. Y. Kim and J. K. Park, Principally quasi-Baer rings, Comm. Algebra, 29(2) (2001), 639-660.
  • G. F. Birkenmeier, J. Y. Kim and J. K. Park, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra, 159(1) (2001), 25-42.
  • G. F. Birkenmeier and J. K. Park, Triangular matrix representations of ring extensions, J. Algebra, 265(2) (2003), 457-477.
  • W. E. Clark, Twisted matrix units semigroup algebras, Duke Math. J., 34(3) (1967), 417-423.
  • N. J. Groenewald, A note on extensions of Baer and p.p.-rings, Publ. Inst. Math. (Beograd) (N.S.), 34(48) (1983), 71-72.
  • Y. Hirano, On ordered monoid rings over a quasi-Baer ring, Comm. Algebra, 29(5) (2001), 2089-2095.
  • I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York- Amsterdam, 1968.
  • A. Khairnar and B. N. Waphare, Order properties of generalized projections, Linear Multilinear Algebra, 65(7) (2017), 1446-1461.
  • B. N. Waphare and A. Khairnar, Semi-Baer modules, J. Algebra Appl., 14(10) (2015), 1550145 (12 pp).
  • Z. Yi and Y. Zhou, Baer and quasi-Baer properties of group rings, J. Aust. Math. Soc., 83(2) (2007), 285-296.
  • L. Zan and J. Chen, p.p. properties of group rings, Int. Electron. J. Algebra, 3 (2008), 117-124.
  • L. Zan and J. Chen, Principally quasi-Baer properties of group rings, Studia Sci. Math. Hungar., 49(4) (2012), 454-465.
There are 14 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Articles
Authors

Anil Khairnar This is me

B. N. Waphare This is me

Publication Date July 11, 2017
Published in Issue Year 2017 Volume: 22 Issue: 22

Cite

APA Khairnar, A., & Waphare, B. N. (2017). Baer Group Rings with Involution. International Electronic Journal of Algebra, 22(22), 1-10. https://doi.org/10.24330/ieja.325913
AMA Khairnar A, Waphare BN. Baer Group Rings with Involution. IEJA. July 2017;22(22):1-10. doi:10.24330/ieja.325913
Chicago Khairnar, Anil, and B. N. Waphare. “Baer Group Rings With Involution”. International Electronic Journal of Algebra 22, no. 22 (July 2017): 1-10. https://doi.org/10.24330/ieja.325913.
EndNote Khairnar A, Waphare BN (July 1, 2017) Baer Group Rings with Involution. International Electronic Journal of Algebra 22 22 1–10.
IEEE A. Khairnar and B. N. Waphare, “Baer Group Rings with Involution”, IEJA, vol. 22, no. 22, pp. 1–10, 2017, doi: 10.24330/ieja.325913.
ISNAD Khairnar, Anil - Waphare, B. N. “Baer Group Rings With Involution”. International Electronic Journal of Algebra 22/22 (July 2017), 1-10. https://doi.org/10.24330/ieja.325913.
JAMA Khairnar A, Waphare BN. Baer Group Rings with Involution. IEJA. 2017;22:1–10.
MLA Khairnar, Anil and B. N. Waphare. “Baer Group Rings With Involution”. International Electronic Journal of Algebra, vol. 22, no. 22, 2017, pp. 1-10, doi:10.24330/ieja.325913.
Vancouver Khairnar A, Waphare BN. Baer Group Rings with Involution. IEJA. 2017;22(22):1-10.