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 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 $\\mathfrak{m}_{p}$ 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

\\displaystyle\\mathfrak{m}_{f}\\mathfrak{m}_{g}=\\mathfrak{m}_{fg}

(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 . (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 generators $a$ , $b$ , $c$ and $d$ 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 (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 multiple (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 domain, 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).