J. Aust. Math. Soc. 83 (2007), no. 2, pp. 285–296.

Baer and quasi-Baer properties of group rings

Zhong Yi Yiqiang Zhou
Department of Mathematics
Guangxi Normal University
Guilin, 541004
P.R. China
zyi@mailbox.gxnu.edu.cn
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John's A1C 5S7
Canada
zhou@math.mun.ca
Received 1 November 2005; revised 1 May 2006
Communicated by J. Du

Abstract

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}\in 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}\in 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 R^G to be Baer.

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2000 Mathematics Subject Classification: primary 16S34; secondary 16E50
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