The valuation structure of homomorphic images of Prüfer domains

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 $D$.