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.
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.
Reviewer: David F. Anderson (Knoxville)
MSC:
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 |