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On prime right alternative rings with commutators in the left nucleus

Published online by Cambridge University Press:  17 April 2009

Erwin Kleinfeld  and
Harry F. Smith
Erwin Kleinfeld
Affiliation:
Division of Mathematical SciencesUniversity of Iowa Iowa City, IA 52242United States of America
Harry F. Smith
Affiliation:
Department of Mathematics, Statistics and Computing ScienceUniversity of New England Armidale, NSW 2351, Australia
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Abstract

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A ring is called s–prime if the 2-sided annihilator of a nonzero ideal must be zero. In particular, any simple ring or prime (—1, 1) ring is s–prime. Also, a nonzero s–prime right alternative ring, with characteristic ≠ 2, cannot be right nilpotent. Let R be a right alternative ring with commutators in the left nucleus. Then R is associative in the following cases: (1) R is prime, with characteristic ≠ 2, and has an idempotent e ≠ 1 such that (e, e, R) = 0. (2) R is an algebra over a commutative-associative ring with 1/6, and R is either s–prime, or R is prime and locally (—1,1).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 50 , Issue 2 , October 1994 , pp. 287 - 298
Copyright
Copyright © Australian Mathematical Society 1994

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