Prüfer domain

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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 C form a Prüfer domain. The ring of integer valued polynomials with rational number coefficients is a Prüfer domain, although the ring Z[X] of integer polynomials is not, Template:Harv. 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, Template:Harv.

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 noncompact Riemann surface is a Bézout domain, Template:Harv, 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, Template:Harv has around forty, and Template:Harv 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
  • For any (finitely generated) ideals I, J, K of R, the following distributivity property holds:
  • For any (finitely generated) nonzero 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
Flatness
Integral closure
  • Every overring of R is integrally closed
  • R is integrally closed and there is some positive integer n such that for every a, b in R one has (a,b)n = (an,bn).
  • R is integrally closed and each element of the quotient field K of R is a root of a polynomial in R[x] whose coefficients generate R as an R-module, Template:Harv.

Properties

  • Though ideals of Dedekind domains can all be generated by two elements, for every positive integer n, there are Prüfer domains with finitely generated ideals that cannot be generated by fewer than n elements, Template:Harv. However, finitely generated maximal ideals of Prüfer domains are two-generated, Template:Harv.
  • A finitely generated module M over a Prüfer domain is projective if and only if it is torsion-free. In fact, this property characterizes Prüfer domains.
  • (Gilmer–Hoffmann Theorem) Suppose that R is an integral domain, K its field of fractions, and S is the integral closure of R in K. Then S is a Prüfer domain if and only if every element of K is a root of a polynomial in R[X] at least one of whose coefficients is a unit of R, Template:Harv.
  • A commutative domain is a Dedekind domain if and only if the torsion submodule is a direct summand whenever it is bounded (M is bounded means rM = 0 for some r in R), Template:Harv. Similarly, a commutative domain is a Prüfer domain if and only if the torsion submodule is a direct summand whenever it is finitely generated, Template:Harv.

Generalizations

More generally a Prüfer ring is a commutative ring in which every non-zero finitely generated ideal consisting only of non-zero-divisors is invertible (that is, projective).

A commutative ring is said to be arithmetical if for every maximal ideal m in R, the localization Rm of R at m is a chain ring. With this definition, an arithmetical domain is a Prüfer domain.

Noncommutative right or left semihereditary domains could also be considered as generalizations of Prüfer domains.

See also

References

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