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Proceedings of the American Mathematical Society

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

 

 

On classical quotients of polynomial identity rings with involution


Author: Louis Halle Rowen
Journal: Proc. Amer. Math. Soc. 40 (1973), 23-29
MSC: Primary 16A28
MathSciNet review: 0323822
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Abstract: Let $ (R, \ast )$ denote a ring $ R$ with involution $ ( \ast )$, where ``involution'' means `` $ {\text{anti - automorphism of order}} \leqq {\text{ two}}$". We can specialize many ring-theoretical concepts to rings with involution; in particular an ideal of $ (R, \ast )$ is an ideal of $ R$ stable under $ ( \ast )$, and the center of $ (R, \ast )$ is the set of central elements of $ R$ which are fixed under $ ( \ast )$. Then we say $ (R, \ast )$ is prime when the product of any two nonzero ideals of $ (R, \ast )$ is nonzero; similarly $ (R, \ast )$ is semiprime when any power of a nonzero ideal of $ (R, \ast )$ is nonzero. The main result of this paper is a strong analogue to Posner's theorem [5], namely that any prime $ (R, \ast )$ with polynomial identity has a ring of quotients $ {R_T}$, formed merely by adjoining inverses of nonzero elements of the center of $ (R, \ast )$. This quotient ring $ ({R_T}, \ast )$ is simple and finite dimensional over its center. An extension of these results to semiprime Goldie rings with polynomial identity is given.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0323822-8
Keywords: Classical ring of quotients, Goldie ring, involution, polynomial identity, prime, semiprime, simple, center, central quotients
Article copyright: © Copyright 1973 American Mathematical Society