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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bond invariance of $ G$-rings and localization


Author: Robert B. Warfield
Journal: Proc. Amer. Math. Soc. 111 (1991), 13-18
MSC: Primary 16N60; Secondary 16D20, 16D30, 16D60, 16P50
MathSciNet review: 1027102
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Abstract: It is proved that if $ R$ and $ S$ are prime Noetherian rings and there exists an $ (R,S)$-bimodule that is finitely generated and torsionfree on each side, then the intersection of the nonzero prime ideals of $ R$ is nonzero if and only if the same holds for the corresponding intersection in $ S$. Consequently, if the right primitive ideals in a given Noetherian ring are precisely the locally closed prime ideals, then the same equivalence holds true for any finite extension ring. Another consequence of the methods used here is the following answer to a question of Braun: If the intersection of the prime ideals in a clique in a Noetherian PI ring is a prime ideal $ Q$, then $ Q$ is localizable.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1027102-8
PII: S 0002-9939(1991)1027102-8
Keywords: Noetherian ring, Noetherian bimodule, bond, prime ideal, primitive ideal, $ G$-ring, locally closed, ring extension, clique, localization, PI ring
Article copyright: © Copyright 1991 American Mathematical Society