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Manis valuations and Prüfer extensions. I: A new chapter in commutative algebra. (English) Zbl 1033.13001
Lecture Notes in Mathematics 1791. Berlin: Springer (ISBN 3-540-43951-X/pbk). 267 p. (2002).
These notes give a careful study of relative Prüfer rings and Manis valuations with an eye towards applications to real and \(p\)-adic algebraic geometry. The main topic is Prüfer ring extensions, where an extension \(A\subset R\) of commutative rings is called Prüfer if \(A\) is \(R\)-Prüfer in the sense of M. Griffin that \((A_{[P]}, P_{[P]})\) is a Manis pair in \(R\) for every maximal ideal \(P\) of \(A\).
This volume has three chapters. The first chapter develops the basic properties of Manis valuations and Prüfer extensions; the second chapter studies Prüfer extensions from a ideal-theoretic rather than a valuation point of view; and the final chapter studies several special types of Manis valuations. As indicated by the title, a second volume is also planned.

13A18 Valuations and their generalizations for commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13A15 Ideals and multiplicative ideal theory in commutative rings
13B02 Extension theory of commutative rings
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
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