# two-generator property

Theorem. Every ideal of a Dedekind domain^{} can be generated by two of its elements.

Proof. Let $\U0001d51e$ be an arbitrary ideal of a Dedekind domain $R$. Let $\U0001d51f$ be such an ideal of $R$ that $\U0001d51e\U0001d51f$ is a principal ideal^{} $(\beta )$. The lemma to which this entry is attached gives also an element $\gamma $ and an ideal $\U0001d520$ of $R$ such that $\U0001d51e\U0001d520=(\gamma )$ and $\U0001d51f+\U0001d520=R$. Then we have

$$\U0001d51e=\mathrm{gcd}(\U0001d51e\U0001d51f,\U0001d51e\U0001d520)=\mathrm{gcd}((\beta ),(\gamma ))=(\beta ,\gamma )$$ |

because $\mathrm{gcd}(\U0001d51f,\U0001d520)=\U0001d51f+\U0001d520=R=(1)$. $\mathrm{\square}$

The Dedekind domains are trivially Prüfer domains, but the two-generator property can not be generalized to the invertible ideals of all Prüfer domains (and Prüfer rings): Schülting has constructed an invertible ideal of a Prüfer domain that can not be generated by less than three generators. The example of Schülting is the fractional ideal^{} $(1,X,Y)$ of the Prüfer domain ${\bigcap}_{j}{B}_{j}$ where the ${B}_{j}$’s run all valuation rings^{} of the rational function field^{} $\mathbb{R}(X,Y)$ which have the residue fields^{} formally real.

## References

- 1 Eben Matlis: “The two-generator problem for ideals”. – The Michigan Mathematical Journal 17 ${\text{N}}^{\circ}$ 3 (1970).
- 2 Heinz-Werner Schülting: “Über die Erzeugendenanzahl invertierbarer Ideale in Prüferringen”. – Communications in Algebra 7 ${\text{N}}^{\circ}$ 13 (1979). [Zentralblatt 432.13010]

Title | two-generator property |
---|---|

Canonical name | TwogeneratorProperty |

Date of creation | 2015-05-05 15:25:37 |

Last modified on | 2015-05-05 15:25:37 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 38 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 11R04 |

Synonym | Ideal of Dedekind domain |

Related topic | SumOfIdeals |

Related topic | FamousAndInfamousOpenQuestionsInMathematics |

Related topic | AnyDivisorIsGcdOfTwoPrincipalDivisors |