Integral Domains whose Simple Overrings are Intersections of Localizations

Details Integral Domains whose Simple Overrings are Intersections of Localizations Integral Domains whose Simple Overrings are Intersections of Localizations

2004-06-15

arxiv.org/abs/math/0406295v1

Mathematics

Commutative Algebra

Scientific paper

Call a domain $R$ an sQQR-domain if each simple overring of $R$, i.e., each ring of the form $R[u]$ with $u$ in the quotient field of $R$, is an intersection of localizations of $R$. We characterize Pr\"ufer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We also show how to construct sQQR-domains which have (non-simple) overrings which are not intersections of localizations.

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