Prüfer domain

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.

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, and several new ones have been discovered since the book was written.

The following conditions on an integral domain "R" are equivalent:
* "R" is a Prüfer domain, i.e. every finitely generated ideal of "R" is projective.
* Every non-zero finitely generated ideal "I" of "R" is invertible: i.e. $I\; cdot\; I^\{-1\}\; =\; R$, where $I^\{-1\}\; =\; \{rin\; q(R):\; rIsubseteq\; R\}$ and $q(R)$ is the field of fractions of "R".

* For any ideals "I", "J", "K" of "R", the following distributivity property holds:

:: $I(Jcap\; K)=IJcap\; IK.$
* For every prime ideal "P" of "R", the localization "R"_"P" of "R" at "P" is a valuation domain.
* For every maximal ideal "m" in "R", the localization "R"_"m" of "R" at "m" is a valuation domain.
* Every torsion-free "R"-module is flat.
* Every torsionless "R"-module is flat.
* Every ideal of "R" is 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.

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).

Properties

* A commutative ring is a Dedekind domain if and only if it is a Prüfer domain and Noetherian.

* If "R" is a Prüfer domain and "K" is its field of fractions, then any ring "S" such that "R" ⊆ "S" ⊆ "K" is a Prüfer domain.

* 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".

References

* cite book
last = Bourbaki
first = Nicolas
authorlink= Nicolas Bourbaki
title = Commutative algebra. Chapters 1–7
series = Elements of Mathematics (Berlin)
language = English, translated from the French, reprint of the 1989 English translation
publisher = Springer-Verlag
location = Berlin
year = 1998
isbn = 3-540-64239-0
* cite journal
author = Gilmer, Robert
coauthors = Hoffmann, Joseph F.
title = A characterization of Prüfer domains in terms of polynomials
journal = Pacific J. Math.
volume = 60
year = 1975
issue = 1
pages = 81–85
issn = 0030-8730
* cite book
last = Lam
first = T. Y.
title = Lectures on modules and rings
series = Graduate Texts in Mathematics No. 189
publisher = Springer-Verlag
location = New York
year = 1999
isbn = 0-387-98428-3