Prüfer ring

Definition.  A commutative ring R with non-zero unity is a Prüfer ring (cf. Prüfer domain) if every finitely generatedMathworldPlanetmathPlanetmathPlanetmath regular ideal of R is invertible. (It can be proved that if every ideal of R generated by two elements is invertible, then all finitely generated ideals are invertible; cf. invertibility of regularly generated ideal.)

Denote generally by  𝔪p  the R-module generated by the coefficientsMathworldPlanetmath of a polynomialMathworldPlanetmathPlanetmath p in T[x], where T is the total ring of fractionsMathworldPlanetmath of R.  Such coefficient modules are, of course, fractional idealsMathworldPlanetmath of R.

Theorem 1 (Pahikkala 1982).  Let R be a commutative ring with non-zero unity and let T be the total ring of fractions of R.  Then, R is a Prüfer ring iff the equation

𝔪f𝔪g=𝔪fg (1)

holds whenever f and g belong to the polynomial ring T[x] and at least one of the fractional ideals 𝔪f and 𝔪g is . (See also product of finitely generated ideals.)

Theorem 2 (Pahikkala 1982).   The commutative ring R with non-zero unity is Prüfer ring iff the multiplication rule


for the integral ideals of R holds whenever at least one of the generatorsPlanetmathPlanetmathPlanetmath a, b, c and d is not zero divisorMathworldPlanetmath.

The proofs are found in the paper

J. Pahikkala 1982: “Some formulae for multiplying and inverting ideals”.  – Annales universitatis turkuensis 183. Turun yliopisto (University of Turku).

Cf. the entries “multiplication rule gives inverse ideal (” and “two-generator property (”.

An additional characterization of Prüfer ring is found here in the entry “least common multipleMathworldPlanetmathPlanetmath (”, several other characterizations in [1] (p. 238–239).

Note.  A commutative ring R satisfying the equation (1) for all polynomials f,g is called a Gaussian ring.  Thus any Prüfer domain ( is always a Gaussian ring, and conversely (, an integral domainMathworldPlanetmath, which is a Gaussian ring, is a Prüfer domain.  Cf. [2].


  • 1 M. Larsen & P. McCarthy: Multiplicative theory of ideals.  Academic Press. New York (1971).
  • 2 Sarah Glaz: “The weak dimensions of Gaussian rings”. – Proc. Amer. Math. Soc. (2005).
Title Prüfer ring
Canonical name PruferRing
Date of creation 2015-05-05 15:21:07
Last modified on 2015-05-05 15:21:07
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 89
Author pahio (2872)
Entry type Theorem
Classification msc 13C13
Classification msc 13F05
Related topic LeastCommonMultiple
Related topic GeneratorsOfInverseIdeal
Related topic ProductOfIdeals
Related topic MultiplicationRing
Related topic PruferDomain
Related topic InvertibilityOfRegularlyGeneratedIdeal
Related topic MultiplicationRuleGivesInverseIdeal
Related topic ContentOfPolynomial
Related topic ProductOfFinitelyGeneratedIdeals
Defines Prüfer ring
Defines coefficient module
Defines Gaussian ring