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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
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$.

References [Enhancements On Off] (What's this?)

  • [1] Monte B. Boisen, Jr. and Max D. Larsen, On Prüfer rings as images of Prüfer domains, Proc. Amer. Math. Soc. 40 (1973), 87-90. MR 0319979 (47:8520)
  • [2] Robert Gilmer, Multiplicative ideal theory, Dekker, New York, 1972. MR 0427289 (55:323)
  • [3] Malcolm Griffin, Prüfer rings with zero divisors, J. Reine Angew. Math. 239/240 (1969), 55-67. MR 41 #188. MR 0255527 (41:188)
  • [4] Max D. Larsen and Paul J. McCarthy, Multiplicative theory of ideals, Academic Press, New York, 1971. MR 0414528 (54:2629)
  • [5] Merle E. Manis, Extension of valuation theory, Bull. Amer. Math. Soc. 73 (1967), 735-736. MR 36 #1436. MR 0218349 (36:1436)

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Keywords: Valuation, valuation pair, Prüfer rings
Article copyright: © Copyright 1974 American Mathematical Society

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