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Difference between revisions of "Prüfer domain"

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A commutative semi-hereditary integral domain (cf. also Semi-hereditary ring; Injective module). The Prüfer domains are the commutative integral domains that satisfy the following two conditions:

i) the localization (cf. also Localization in a commutative algebra) is a valuation ring (see Valuation) for every maximal ideal of ;

ii) an -module is flat (cf. Flat module) if and only if it is torsion-free (cf. also Group without torsion).

A Noetherian Prüfer domain is a Dedekind domain (see Dedekind ring). Each Bezout domain (also called 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 [a2] (hence, these are not Dedekind rings).

References

[a1] R. Gilmer, "Multiplicative ideal theory" , M. Dekker (1972)
[a2] R. Heidman, L. Levy, " and -generator ideals in Prüfer domains" Rocky Mount. Math. J. , 5 (1975) pp. 361–373
[a3] H.C. Hutchins, "Examples of commutative rings" , Polygonal (1981)
How to Cite This Entry:
Prüfer domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pr%C3%BCfer_domain&oldid=23504
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article