Acta Mathematica Vietnamica
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Extensions of Valuation Domains and Going-up
Shai Sarussipp. 683-687
Abstract
Suppose F is a field with valuation v and valuation domain $O_v$, E/F is a finite-dimensional field extension, and R is an $O_v$-subalgebra of E such that F ⋅ R = E and R ∩ F =$O_v$. It is known that R satisfies LO, INC, GD and SGB over $O_v$; it is also known that under certain conditions R satisfies GU over $O_v$. In this paper, we present a necessary and sufficient condition for the existence of such R that does not satisfy GU over $O_v$. We also present an explicit example of such R that does not satisfy GU over $O_v$.