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In [[mathematics]], a '''Prüfer domain''' is a type of [[commutative ring]] that generalizes [[Dedekind domain]]s in a non-[[Noetherian ring|Noetherian]] context.  These rings possess the nice [[ideal (ring theory)|ideal]] and [[module (mathematics)|module]] theoretic properties of Dedekind domains, but usually only for [[finitely generated module]]s.  Prüfer domains are named after the [[Germany|German]] [[mathematician]] [[Heinz Prüfer]].

== Examples ==

The ring of [[entire function]]s on the open complex plane '''C''' form a Prüfer domain.  The ring of [[integer valued polynomial]]s with [[rational number]] coefficients is a Prüfer domain.  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 the [[direct limit]] of Prüfer domains is a Prüfer domain, {{harv|Fuchs|Salce|2001|pp=93–94}}.

Many Prüfer domains are also [[Bézout domain]]s, that is, not only are finitely generated ideals [[projective module|projective]], they are even [[free module|free]] (that is, [[principal ideal|principal]]).  For instance the ring of analytic functions on any noncompact [[Riemann surface]] is a Bézout domain, {{harv|Helmer|1940}}, and the ring of algebraic integers is Bézout.

== Definitions ==

A '''Prüfer domain''' is a [[semihereditary ring|semihereditary]]  [[integral domain]]. Equivalently, a Prüfer domain may be defined as a [[commutative ring]] without [[zero divisor]]s in which every non-zero [[finitely generated module|finitely generated]] [[ideal (ring theory)|ideal]] is invertible.  Many different characterizations of Prüfer domains are known. Bourbaki lists fourteen of them, {{harv|Gilmer|1972}} has around forty, and {{harv|Fontana|Huckaba|Papick|1997|p=2}} 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 module|projective]]:
;Ideal arithmetic:
* Every non-zero finitely generated ideal ''I'' of ''R'' is '''invertible''': i.e. <math>\ I \cdot I^{-1} =  R</math>, where <math>I^{-1} = \{r\in q(R): rI\subseteq R\}</math> and <math>\ q(R)</math> is the [[field of fractions]] of ''R''.<!-- {{harv|Fontana|Huckaba|Papick|1997|p=2}}, 2 -->  Equivalently, every non-zero ideal generated by two elements is invertible.<!-- {{harv|Fontana|Huckaba|Papick|1997|p=2}}, 3 -->
* For any (finitely generated) nonzero ideals ''I'', ''J'', ''K'' of ''R'', the following distributivity property holds:
:: <math> I \cap (J + K) = (I \cap J) + (I \cap K). </math> <!-- {{harv|Gilmer|1972|p=310}} b -->
* For any (finitely generated) ideals ''I'', ''J'', ''K'' of ''R'', the following distributivity property holds:
:: <math> I(J \cap K)=IJ \cap IK. </math> <!-- {{harv|Gilmer|1972|p=310}} c -->
* For any (finitely generated) nonzero ideals ''I'', ''J'' of ''R'', the following property holds:
:: <math> (I+J)(I \cap J) = IJ. </math> <!-- {{harv|Gilmer|1972|p=310}} d -->
*For any finitely generated ideals ''I'', ''J'', ''K'' of ''R'', if ''IJ'' = ''IK'' then ''J'' = ''K'' or ''I'' = 0. <!-- {{harv|Gilmer|1972|p=298}} -->
<!-- {{harv|Gilmer|1972|p=310}} e,f,g not included, deals with quotients/colon ideals -->
;Localizations:
* For every [[prime ideal]] ''P'' of ''R'', the [[localization of a ring|localization]] ''R''<sub>''P''</sub> of ''R'' at ''P'' is a [[valuation ring|valuation domain]].<!-- {{harv|Fontana|Huckaba|Papick|1997|p=2}}, 1a -->
* For every [[maximal ideal]] ''m'' in ''R'', the localization ''R''<sub>''m''</sub> of ''R'' at ''m'' is a valuation domain.<!-- {{harv|Fontana|Huckaba|Papick|1997|p=2}}, 1b -->
* ''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 (algebra)|torsion-free]] ''R''-module is [[flat module|flat]].
* Every [[torsionless module|torsionless]] ''R''-module is flat.
* Every ideal of ''R'' is flat
* Every overring of ''R'' is ''R''-flat <!-- {{harv|Fontana|Huckaba|Papick|1997|p=2}}, 5 -->
* Every submodule of a flat ''R''-module is flat.
* If ''M'' and ''N'' are torsion-free ''R''-modules then their [[tensor product]] ''M''&nbsp;⊗<sub>''R''</sub>&nbsp;''N'' is torsion-free.
* If ''I'' and ''J'' are two ideals of ''R'' then ''I''&nbsp;⊗<sub>''R''</sub>&nbsp;''J'' is torsion-free.
* The [[torsion submodule]] of every [[finitely generated module]] is a [[direct summand]], {{harv|Kaplansky|1960}}.
;Integral closure:
* Every overring of ''R'' is [[integral closure|integrally closed]]<!-- {{harv|Fontana|Huckaba|Papick|1997|p=2}}, 4 -->
* ''R'' is integrally closed and there is some positive integer ''n'' such that for every ''a'', ''b'' in ''R'' one has (''a'',''b'')<sup>''n''</sup> = (''a''<sup>''n''</sup>,''b''<sup>''n''</sup>). <!-- {{harv|Fontana|Huckaba|Papick|1997|p=2}}, 9 -->
* ''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, {{harv|Gilmer|Hoffman|1975|p=81}}.

== Properties ==

* A commutative ring is a [[Dedekind domain]] if and only if it is a Prüfer domain and [[Noetherian ring|Noetherian]].
* Though Prüfer domains need not be Noetherian, they must be [[coherent ring|coherent]], since finitely generated projective modules are [[finitely generated module|finitely related]].

* 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, {{harv|Swan|1984}}.  However, finitely generated maximal ideals of Prüfer domains are two-generated, {{harv|Fontana|Huckaba|Papick|1997|p=31}}.

* 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.
* If ''R'' is a Prüfer domain, ''K'' is its [[field of fractions]], and ''L'' is an [[algebraic extension field]] of ''K'', then the integral closure of ''R'' in ''L'' is a Prüfer domain, {{harv|Fuchs|Salce|2001|p=93}}.

* A finitely generated [[module (ring theory)|module]] ''M'' over a Prüfer domain is [[projective module|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 (ring theory)|unit]] of ''R'', {{harv|Gilmer|Hoffman|1975|loc=Theorem 2}}.

* 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''), {{harv|Chase|1960}}.  Similarly, a commutative domain is a Prüfer domain if and only if the torsion submodule is a direct summand whenever it is finitely generated, {{harv|Kaplansky|1960}}.

== Generalizations ==

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

A commutative ring is said to be [[arithmetical ring|arithmetical]] if for every [[maximal ideal]] ''m'' in ''R'', the localization ''R''<sub>''m''</sub> 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 ==
*[[Divided domain]]

==References==
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{{DEFAULTSORT:Prufer domain}}
[[Category:Commutative algebra]]