Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T17:13:20.585Z Has data issue: false hasContentIssue false
  • English
  • Français

Baer and quasi-Baer properties of group rings

Part of: Rings and algebras arising under various constructions Homological methods

Published online by Cambridge University Press:  09 April 2009

Zhong Yi  and
Yiqiang Zhou
Zhong Yi
Affiliation:
Department of Mathematics, Guangxi Normal University, Guilin, 541004, P.R. [email protected]
Yiqiang Zhou
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's A1C 5S7, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A ring R is said to be a Baer (respectively, quasi-Baer) ring if the left annihilator of any nonempty subset (respectively, any ideal) of R is generated by an idempotent. It is first proved that for a ring R and a group G, if a group ring RG is (quasi-) Baer then so is R; if in addition G is finite then |G|–1 € R. Counter examples are then given to answer Hirano's question which asks whether the group ring RG is (quasi-) Baer if R is (quasi-) Baer and G is a finite group with |G|–1 € R. Further, efforts have been made towards answering the question of when the group ring RG of a finite group G is (quasi-) Baer, and various (quasi-) Baer group rings are identified. For the case where G is a group acting on R as automorphisms, some sufficient conditions are given for the fixed ring RG to be Baer.

MSC classification

Secondary: 16S34: Group rings , Laurent polynomial rings 16E50: von Neumann regular rings and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 2 , October 2007 , pp. 285 - 296
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Birkenmeier, G. F., Kim, J. Y. and Park, J. K., ‘On quasi-Baer rings’, in: Algebra and its applications (Athens, OH, 1999), Contemp. Math. 259 (Amer. Math. Soc, Providence, RI, 2000) pp. 67–92.CrossRefGoogle Scholar
[2]Birkenmeier, G. F., Kim, J. Y., ‘Polynomial extensions of Baer and quasi-Baer rings’, J. Pure Appl. Algebra 159 (2001), 25–42.CrossRefGoogle Scholar
[3]Birkenmeier, G.F. and Park, J.K., ‘Triangular matrix representations of ring extensions’, J. Algebra 265 (2003), 457–477.CrossRefGoogle Scholar
[4]Chen, J., Li, Y. and Zhou, Y., ‘Morphic group rings’, J. Pure Appl. Algebra 205 (2006), 621–639.CrossRefGoogle Scholar
[5]Clark, W. E., ‘Twisted matrix units semigroup algebras’, Duke Math. J. 34 (1967), 417–424.CrossRefGoogle Scholar
[6]Groenewald, N. J., ‘A note on extensions of Baer and p.p.-rings’, Publ. de L’institut Math. 34 (1983), 71–72.Google Scholar
[7]Hirano, Y., ‘On ordered monoid rings over a quasi-Baer ring’, Comm. Algebra 29 (2001), 2089–2095.CrossRefGoogle Scholar
[8]Hirano, Y., ‘Open problems’, in: Proceedings of the 3rd Korea-China-Japan Symposium and the 2nd Korea-Japan Joint Seminar held in Kyongju, June 28–July 3, 1999, Trends Math. (2001) pp. 442.Google Scholar
[9]Kaplansky, I., Rings of Operators, Math. Lecture Notes Series (Benjamin, New York, 1965).Google Scholar
[10]McConnell, J. C. and Robson, J. C., Noncommutative Noetherian Rings (John Wiley & Sons, Chichester, 1987).Google Scholar
[11]Passman, D., The Algebraic Structure of Group Rings (John Wiley & Sons, New York-London-Sydney, 1977).Google Scholar
[12]Pollingher, A. and Zaks, A., ‘On Baer and quasi-Baer rings’, Duke Math. J. 37 (1970), 127–138.CrossRefGoogle Scholar
[13]Yi, Z., ‘Homological dimension of skew group rings and crossed products’, J. Algebra 164 (1994), 101–123.CrossRefGoogle Scholar