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Unitaries in Simple Artinian Rings

Published online by Cambridge University Press:  20 November 2018

M. Chacron
M. Chacron*
Affiliation:
Carleton University, Ottawa, Ontario
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Let R be a 2-torsion free simple artinian ring with involution*. The element u of R is said to be unitary if u is invertible with inverse u*. In this paper we shall be concerned with the subalgebras W of R over its centre Z such that uWu* ⊆ W, for all unitaries u of R. We prove that if R has rank superior to 1 over a division ring D containing more than 5 elements and if R is not 4-dimensional then any such subalgebra W must be one of the trivial subalgebras 0, Z or R, under one of the following extra finiteness assumptions: W contains inverses, W satisfies a polynomial identity, the ground division ring D is algebraic, the involution is a conjugate-transpose involution such that D equipped with the induced involution is generated by unitaries.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 3 , 01 June 1979 , pp. 542 - 557
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Chacron, M., Lie action of certain skews in *-rings, Can. J. Math, (to appear).Google Scholar
2. Handelman, D., Involutory division algebras not generated by their unitaries (manuscript).Google Scholar
3. Herstein, I.N., Lectures on rings with involution (University of Chicago Press, Chicago, 1976).Google Scholar
4. Lanski, C., Unitary invariance in algebraic algebras (submitted).Google Scholar
5. Montgomery, S. and McCrimmon, K., Open questions on rings with involution. Ring Theory Conference, University of Chicago, July, 1973.Google Scholar