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Proceedings of the American Mathematical Society

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

 

 

Separation of nonassociates by valuations


Authors: David E. Brown and Max D. Larsen
Journal: Proc. Amer. Math. Soc. 31 (1972), 326-332
DOI: https://doi.org/10.1090/S0002-9939-1972-0289506-9
MathSciNet review: 0289506
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Abstract: In many classical integral domains, given two nonassociates it is possible to find a valuation on the quotient field of the domain which is nonnegative on the domain and for which the nonassociates have different values. Recent work by Griffin, Harrison, and Manis has extended valuation theory to commutative rings with identity which contain zero divisors. In this paper we investigate the separation of nonassociates by valuations for the extended valuation theory. Our main result states that if $ R$ is a ring with a von Neumann regular total quotient ring, then nonassociates can be separated by valuations if and only if there is no unit in the integral closure of $ R$ which is not a unit in $ R$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0289506-9
Keywords: Separation of nonassociates by valuations, large Jacobson radical, $ R$ is close to $ S$
Article copyright: © Copyright 1972 American Mathematical Society