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Proceedings of the American Mathematical Society

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

 

 

Localization in a principal right ideal domain


Author: Raymond A. Beauregard
Journal: Proc. Amer. Math. Soc. 31 (1972), 21-23
DOI: https://doi.org/10.1090/S0002-9939-1972-0291198-X
MathSciNet review: 0291198
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Abstract | References | Additional Information

Abstract: Let $ R$ be a principal right ideal domain with right $ D$-chain $ \{ {R^{(\alpha )}}\vert \leqq \alpha \leqq \delta \} $, and let $ {K_\alpha } = R{({R^{(\alpha )}})^{ - 1}}$ be the associated chain of quotient rings of $ R$. The local skew degree of $ R$ is defined to be the least ordinal $ \lambda $ such that $ {K_\lambda }$ is a local ring. The main result states that for each $ \alpha \geqq \lambda ,{K_\alpha }$ is a local ring; equivalently, $ R$ has a unique $ (\alpha + 1)$-prime for $ \delta > \alpha \geqq \lambda $.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0291198-X
Keywords: PRI domain, $ D$-chain, primes, infinite primes, localization
Article copyright: © Copyright 1972 American Mathematical Society