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Unitaries in Simple Artinian Rings

Published online by Cambridge University Press:  20 November 2018

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
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Chacron, M., Lie action of certain skews in *-rings, Can. J. Math, (to appear).Google Scholar
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5. Montgomery, S. and McCrimmon, K., Open questions on rings with involution. Ring Theory Conference, University of Chicago, July, 1973.Google Scholar