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1	Polynomial Ring
1	Proof
1	Biconditional
1	Integral Domain
1	Commutative Ring
1	Unity
1	Ideal
1	Multiplication
1	Generating Set Of A Group
1	Equation
2	Characterization
3	Coefficients Of A Polynomial
3	Fractional Ideal Of Commutative Ring
3	Zero Divisor
3	Presentationgroup
4	Finitely Generated R Module
5	Fractional Ideal
6	Regular Ideal
6	Least Common Multiple
8	Total Ring Of Fractions
10	Prufer Domain
14	Two Generator Property
17	Product Of Finitely Generated Ideals
17	Invertibility Of Regularly Generated Ideal
18	Multiplication Rule Gives Inverse Ideal

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Prufer Ring

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

Denote generally by $\mathfrak{m}_p$ the -module generated by the coefficients of a polynomial in , where is the total ring of fractions of . Such coefficient modules are, of course, fractional ideals of .

Theorem 1 (Pahikkala 1982) Let

be a commutative ring with non-zero unity and let

be the total ring of fractions of

. Then,

is a Prüfer ring iff the equation

$\displaystyle \mathfrak{m}_f\mathfrak{m}_g = \mathfrak{m}_{fg}$

(1)

holds whenever

and

belong to the polynomial ring

and at least one of the fractional ideals $\mathfrak{m}_f$ and $\mathfrak{m}_g$ is regular. (See also product of finitely generated ideals.)

Theorem 2 (Pahikkala 1982) The commutative ring

with non-zero unity is Prüfer ring iff the multiplication rule

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

for the integral ideals of

holds whenever at least one of the generators

and

is not zero divisor.

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 multiple”, several other characterizations in [1] (p. 238-239).

Note. A commutative ring 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 domain, which is a Gaussian ring, is a Prüfer domain. Cf. [2].

Bibliography

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).