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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A generalization of a theorem of Jacobson

Author: Susan Montgomery
Journal: Proc. Amer. Math. Soc. 28 (1971), 366-370
MSC: Primary 16.58
MathSciNet review: 0276272
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Abstract: A well-known theorem of Jacobson asserts that a ring $ R$ in which $ {x^{n(x)}} = x$ for each $ x$ in $ R$ must be commutative. This paper gives a description of a ring with involution in which the above condition is imposed only on the symmetric elements. In particular, if $ R$ is primitive, $ R$ is either commutative or the $ 2 \times 2$ matrices over a field, and, in general, any such $ R$ is locally finite and satisfies a polynomial identity of degree 8.

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Keywords: Rings with involution, commutativity, polynomial identity, algebraic algebras
Article copyright: © Copyright 1971 American Mathematical Society