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单词 PruferRing
释义

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

(a,b)(c,d)=(ac,ad+bc,bd)

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 (http://planetmath.org/MultiplicationRuleGivesInverseIdeal)” and “two-generator property (http://planetmath.org/TwoGeneratorProperty)”.

An additional characterization of Prüfer ring is found here in the entry “least common multipleMathworldPlanetmathPlanetmath (http://planetmath.org/LeastCommonMultiple)”, 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 (http://planetmath.org/PruferDomain) is always a Gaussian ring, and conversely (http://planetmath.org/Converse), an integral domainMathworldPlanetmath, which is a Gaussian ring, is a Prüfer domain.  Cf. [2].

References

  • 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).
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