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Complementation of the Jacobson group in a matrix ring

Published online by Cambridge University Press:  17 April 2009

David Dolžan
David Dolžan
Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19, Ljubljana 1000, Slovenia
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The Jacobson group of a ring R (denoted by = (R)) is the normal subgroup of the group of units of R (denoted by G(R)) obtained by adding 1 to the Jacobson radical of R (J(R)). Coleman and Easdown in 2000 showed that the Jacobson group is complemented in the group of units of any finite commutative ring and also in the group of units a n × n matrix ring over integers modulo ps, when n = 2 and p = 2, 3, but it is not complemented when p ≥ 5. In 2004 Wilcox showed that the answer is positive also for n = 3 and p = 2, and negative in all the remaining cases. In this paper we offer a different proof for Wilcox's results and also generalise the results to a matrix ring over an arbitrary finite commutative ring. We show this by studying the generators and relations that define a matrix ring over a field. We then proceed to examine the complementation of the Jacobson group in the matrix rings over graded rings and prove that complementation depends only on the 0-th grade.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 2 , October 2005 , pp. 317 - 324
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Coleman, C. and Easdown, D., ‘Complementtionin the group of units of a ring’, Bull. Austral. Math. Socl. 62 (2000), 183–192.CrossRefGoogle Scholar
[2]Hall, M., The theory of groups (Macmillan, New York, 1959).Google Scholar
[3]Karpilovsky, G., Group representations, Vol 1 (Elsevier Science Publishers B.V., Amsterdam, 1992).Google Scholar
[4]McDonald, B.R., Finite rings with identity (Marcel Dekker Inc., New York).Google Scholar
[5]Raghavendran, R., ‘Finite associative rings’, Compositio Math. 21 (1969), 195–229.Google Scholar
[6]Rotman, J.J., The theory of groups, an introduction (Allyn and Bacon, Inc., Boston, 1971).Google Scholar
[7]Wilcox, S., ‘Complementation in the group of units of matrix rings’, Bull. Austral. Math. Soc. 70 (2004), 223–227.CrossRefGoogle Scholar