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Invariant Subrings in Rings with Involution

Published online by Cambridge University Press:  20 November 2018

Charles Lanski
Charles Lanski*
Affiliation:
University of Southern California, Los Angeles, California
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The purpose of this paper is to consider, in rings with involution, the structure of those subrings which are invariant under Lie commutation with [K, K]. Our goal is to find conditions which force such subrings to contain a noncentral ideal of the ring. Of course, the subring itself may lie in the center. Orders in 4 × 4 matrix rings over fields are known to provide examples of invariant subrings which are not central and contain no ideal (see [1, 40] or [5]). Except for subdirect products of these two kinds of “counter-examples”, we show that in semi-prime rings, invariant subrings do contain noncentral ideals. This generalizes work of Herstein [4] in two directions by considering semi-prime rings rather than simple rings, and by using [K, K] instead of K.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 1 , 01 February 1978 , pp. 85 - 94
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Herstein, I. N., Topics in ring theory (University of Chicago Press, Chicago, 1969).Google Scholar
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4. Herstein, I. N., Certain submodules of simple rings with involution II, Can. J. Math. 27 (1975), 629–635.Google Scholar
5. Lanski, C., Lie structure in semi-prime rings with involution, Comm. in Alg. 4 (1976), 731–746.Google Scholar
6. Lanski, C. and Montgomery, S., Lie structure of prime rings of characteristic 2, Pacific J. Math. 42 (1972), 117–136.Google Scholar