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Ordered algebraic structures --- Algebraic geometry

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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.

Commutative algebra. --- Commutative rings. --- Prüfer rings. --- Commutative semihereditary domains --- Commutative semihereditary entire rings --- Domains, Commutative semihereditary --- Domains, Prüfer --- Entire rings, Commutative semihereditary --- Prüfer domains --- Prüfer's domains --- Prüfer's rings --- Semihereditary domains, Commutative --- Semihereditary entire rings, Commutative --- Rings (Algebra) --- Algebra --- Algebra. --- Commutative Rings and Algebras. --- Mathematics --- Mathematical analysis

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Forms, Quadratic --- Forms, Pfister --- Algebraic fields --- Quadratic forms --- Diophantine analysis --- Forms, Binary --- Number theory --- Pfister forms --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra) --- Formes quadratiques --- Corps et polynomes