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Proceedings of the American Mathematical Society

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

 

 

On valuation rings that contain zero divisors


Author: James A. Huckaba
Journal: Proc. Amer. Math. Soc. 40 (1973), 9-15
MSC: Primary 13F05
MathSciNet review: 0318134
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Abstract: Let $ R$ be a commutative ring with identity. A new proof is given of the theorem due to Samuel and Griffin which states that $ R$ is integrally closed in its total quotient ring if and only if $ R$ is the intersection of $ B$-valuation rings. We then prove the main result of the paper: If $ K$ is a $ \pi $-regular ring, then $ K$ admits only Prüfer rings as valuation rings.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0318134-2
Keywords: Commutative rings with zero divisors, valuation ring, Prüfer ring, von Neumann regular ring, integral closure of a ring
Article copyright: © Copyright 1973 American Mathematical Society