In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.
Examples
The ring of entire functions on the open complex plane form a Prüfer domain. The ring of integer valued polynomials with rational coefficients is a Prüfer domain, although the ring of integer polynomials is not . While every number ring is a Dedekind domain, their union, the ring of algebraic integers, is a Prüfer domain. Just as a Dedekind domain is locally a discrete valuation ring, a Prüfer domain is locally a valuation ring, so that Prüfer domains act as non-noetherian analogues of Dedekind domains. Indeed, a domain that is the direct limit of subrings that are Prüfer domains is a Prüfer domain .
Many Prüfer domains are also Bézout domains, that is, not only are finitely generated ideals projective, they are even free (that is, principal). For instance the ring of analytic functions on any non-compact Riemann surface is a Bézout domain , and the ring of algebraic integers is Bézout.
Definitions
A Prüfer domain is a semihereditary integral domain. Equivalently, a Prüfer domain may be defined as a commutative ring without zero divisors in which every non-zero finitely generated ideal is invertible. Many different characterizations of Prüfer domains are known. Bourbaki lists fourteen of them, has around forty, and open with nine.
As a sample, the following conditions on an integral domain R are equivalent to R being a Prüfer domain, i.e. every finitely generated ideal of R is projective: Ideal arithmetic Every non-zero finitely generated ideal I of R is invertible: i.e. , where and is the field of fractions of R. Equivalently, every non-zero ideal generated by two elements is invertible. For any (finitely generated) non-zero ideals I, J, K of R, the following distributivity property holds:
For any (finitely generated) ideals I, J, K of R, the following distributivity property holds:
For any (finitely generated) non-zero ideals I, J of R, the following property holds:
For any finitely generated ideals I, J, K of R, if IJ = IK then J = K or I = 0.
Localizations
For every prime ideal P of R, the localization RP of R at P is a valuation domain. For every maximal ideal m in R, the localization Rm of R at m is a valuation domain. R is integrally closed and every overring of R (that is, a ring contained between R and its field of fractions) is the intersection of localizations of R Flatness Every torsion-free R-module is flat. Every torsionless R-module is flat. Every ideal of R is flat Every overring of R is R-flat Every submodule of a flat R-module is flat. If M and N are torsion-free R-modules then their tensor product M ⊗R N is torsion-free. If I and J are two ideals of R then I ⊗R J is torsion-free. The torsion submodule of every finitely generated module is a direct summand . Integral closure Every overring of is integrally closed is integrally closed and there is some positive integer such that for every , in one has . is integrally closed and each element of the quotient field of is a root of a polynomial in whose coefficients generate as an -module .