02456nam a2200325 i 4500991002951659707536160722s2014 sz b 001 0 eng d978331903211510.1007/978-3-319-03212-2doib14259199-39ule_inst512.4423AMS 13A18AMS 13B02AMS 13F05LC QA251.3Knebusch, Manfred54845Manis valuations and Prüfer extensions II /Manfred Knebusch, Tobias KaiserCham :Springer,2014xii, 190 p. :ill. ;24 cmLecture notes in mathematics,0075-8434 ;2103Overrings 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 ; IndexThis 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 KroneckerAlgebraKaiser, Tobiasauthorhttp://id.loc.gov/vocabulary/relators/aut524807.b1425919911-11-1622-07-16991002951659707536LE013 13A KNE11 (2014)12013000293455le013pE36.39-l- 01010.i1578701111-11-16Manis valuations and Prüfer extensions II820720UNISALENTOle01322-07-16ma -engsz 00