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 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

(1)

holds whenever and belong to the polynomial ring and at least one of the fractional ideals and 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

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 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).