Manis valuations and Prüfer extensions. I

Abstract

We call a commutative ring extension $A\subset R$ Prüfer, if $A$ is an $R$-Prüfer ring in the sense of Griffin (Can. J. Math. 26 (1974)). These extensions relate to Manis valuations in much the same way as Prüfer domains to Krull valuations. We develop a basic theory of Prüfer extensions and give some examples. In the introduction we try to explain why Prüfer extensions deserve interest from a geometric viewpoint.