View source for Prüfer domain

A commutative semi-hereditary [[Integral domain|integral domain]] (cf. also [[Semi-hereditary ring|Semi-hereditary ring]]; [[Injective module|Injective module]]). The Prüfer domains are the commutative integral domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110260/p1102601.png" /> that satisfy the following two conditions:

i) the localization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110260/p1102602.png" /> (cf. also [[Localization in a commutative algebra|Localization in a commutative algebra]]) is a valuation ring (see [[Valuation|Valuation]]) for every maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110260/p1102603.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110260/p1102604.png" />;

ii) an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110260/p1102605.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110260/p1102606.png" /> is flat (cf. [[Flat module|Flat module]]) if and only if it is torsion-free (cf. also [[Group without torsion|Group without torsion]]).

A Noetherian Prüfer domain is a Dedekind domain (see [[Dedekind ring|Dedekind ring]]). Each Bezout domain (also called [[Bezout ring|Bezout ring]]) is a Prüfer domain.

There are Prüfer domains that are not Bezout and there are Prüfer domains having finitely generated ideals requiring more than two generators [[#References|[a2]]] (hence, these are not Dedekind rings).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Gilmer,   "Multiplicative ideal theory" , M. Dekker  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Heidman,   L. Levy,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110260/p1102607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110260/p1102608.png" />-generator ideals in Prüfer domains"  ''Rocky Mount. Math. J.'' , '''5'''  (1975)  pp. 361–373</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.C. Hutchins,   "Examples of commutative rings" , Polygonal  (1981)</TD></TR></table>