# 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 ${\U0001d52a}_{p}$ 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

${\U0001d52a}_{f}{\U0001d52a}_{g}={\U0001d52a}_{fg}$ | (1) |

holds whenever $f$ and $g$ belong to the polynomial ring $T[x]$ and at least one of the fractional ideals ${\U0001d52a}_{f}$ and ${\U0001d52a}_{g}$ 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).

Title | Prüfer ring |

Canonical name | PruferRing |

Date of creation | 2015-05-05 15:21:07 |

Last modified on | 2015-05-05 15:21:07 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 89 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 13C13 |

Classification | msc 13F05 |

Related topic | LeastCommonMultiple |

Related topic | GeneratorsOfInverseIdeal |

Related topic | ProductOfIdeals |

Related topic | MultiplicationRing |

Related topic | PruferDomain |

Related topic | InvertibilityOfRegularlyGeneratedIdeal |

Related topic | MultiplicationRuleGivesInverseIdeal |

Related topic | ContentOfPolynomial |

Related topic | ProductOfFinitelyGeneratedIdeals |

Defines | Prüfer ring |

Defines | coefficient module |

Defines | Gaussian ring |