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Proceedings of the American Mathematical Society

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

 

 

The valuation structure of homomorphic images of Prüfer domains


Authors: Monte B. Boisen and Philip B. Sheldon
Journal: Proc. Amer. Math. Soc. 46 (1974), 335-342
MSC: Primary 13F05
DOI: https://doi.org/10.1090/S0002-9939-1974-0404246-2
MathSciNet review: 0404246
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Abstract: Let $ R$ denote a Prüfer ring which is a homomorphic image of a Prüfer domain $ D$. The purpose of this paper is to investigate the relationship between the valuation structure over $ D$ and the valuation structure over $ R$. It is shown that there exists a one-to-one correspondence between the valuations over $ R$ and the valuations over $ D$ centered on primes containing the kernel of the homomorphism. This correspondence is shown to be natural in the sense that the value of an element of $ R$ is either infinity or the value of its pre-images under the corresponding valuation over $ D$. Moreover, the value group of a valuation over $ R$ is an isolated subgroup of the value group of the corresponding valuation over $ D$.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0404246-2
Keywords: Valuation, valuation pair, Prüfer rings
Article copyright: © Copyright 1974 American Mathematical Society