Prüfer ring

Definition. A commutative ring $R$ with non-zero unity is a Prüfer ring (cf. Prüfer domain) if every finitely generated regular ideal of $R$ is invertible. (It can be proved that if every regular ideal of $R$ 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 $R$ -module generated by the coefficients of a polynomial $p$ in $T[x]$ , where $T$ is the total ring of fractions of $R$ . Such coefficient modules are, of course, fractional ideals 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

(1)

holds whenever $f$ and $g$ belong to the polynomial ring $T[x]$ and at least one of the fractional ideals $\mathfrak{m}_f$ and $\mathfrak{m}_g$ is regular. (See also product of finitely generated ideals.)

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 $R$ 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 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).