Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T00:05:18.018Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

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.

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
2. Herstein, I. N., On the Lie structure of an associative ring, J. Algebra 14 (1970), 561–571.Google Scholar
3. Herstein, I. N., A unitary version of the Brauer-Cartan-Hua theorem, J. Algebra 32 (1974), 554–560.Google Scholar
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