Author	Knebusch, Manfred. author
Title	Manis Valuations and Prüfer Extensions II [electronic resource] / by Manfred Knebusch, Tobias Kaiser
Imprint	Cham : Springer International Publishing : Imprint: Springer, 2014
Connect to	http://dx.doi.org/10.1007/978-3-319-03212-2
Descript	XII, 190 p. online resource

This volume is a sequel to “Manis Valuation and Prüfer Extensions I,” LNM1791. The Prüfer extensions of a commutative ring A are roughly those commutative ring extensions R / A,where commutative algebra is governed by Manis valuations on R with integral values on A. These valuations then turn out to belong to the particularly amenable subclass of PM (=Prüfer-Manis) valuations. While in Volume I Prüfer extensions in general and individual PM valuations were studied, now the focus is on families of PM valuations. One highlight is the presentation of a very general and deep approximation theorem for PM valuations, going back to Joachim Gräter’s work in 1980, a far-reaching extension of the classical weak approximation theorem in arithmetic. Another highlight is a theory of so called “Kronecker extensions,” where PM valuations are put to use in  arbitrary commutative  ring extensions in a way that ultimately goes back to the work of Leopold Kronecker

Overrings and PM-Spectra -- Approximation Theorems -- Kronecker extensions and star operations -- Basics on Manis valuations and Prufer extensions -- Multiplicative ideal theory -- PM-valuations and valuations of weaker type -- Overrings and PM-Spectra -- Approximation Theorems -- Kronecker extensions and star operations -- Appendix -- References -- Index

1. Mathematics
2. Commutative algebra
3. Commutative rings
4. Mathematics
5. Commutative Rings and Algebras