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.
