Manis Valuations and Prüfer Extensions II, 2014
Lecture Notes in Mathematics Series, Vol. 2103

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.

Includes supplementary material: sn.pub/extras