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[parent] two-generator property (Theorem)
Theorem 1   Every ideal of a Dedekind domain can be generated by two of its elements.

Proof. Let $\mathfrak{a}$ be an arbitrary ideal of a Dedekind domain $R$ . Let $\mathfrak{b}$ be such an ideal of $R$ that $\mathfrak{ab}$ is a principal ideal $(\beta)$ . The lemma to which this entry is attached gives also an element $\gamma$ and an ideal $\mathfrak{c}$ of $R$ such that $\mathfrak{ac} = (\gamma)$ and $\mathfrak{b+c} = R$ . Then we have $$\mathfrak{a} = \gcd(\mathfrak{ab},\,\mathfrak{ac}) = \gcd((\beta),\,(\gamma)) = (\beta,\,\gamma)$$ because $\gcd(\mathfrak{b},\,\mathfrak{c}) = \mathfrak{b+c} = R = (1)$ . $ \Box$

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.

Bibliography

1
EBEN MATLIS: ``The two-generator problem for ideals''. - The Michigan Mathematical Journal 17 $\mbox{N}\sp\circ$ 3 (1970).
2
HEINZ-WERNER SCHÜLTING: ``Über die Erzeugendenanzahl invertierbarer Ideale in Prüferringen''. - Communications in Algebra 7 $\mbox{N}\sp\circ$ 13 (1979). [Zentralblatt 432.13010]




"two-generator property" is owned by pahio. [ full author list (2) | owner history (2) ]
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See Also: sum of ideals, famous open questions in mathematics, any divisor is gcd of two principal divisors

Other names:  Ideal of Dedekind domain

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invertibility of regularly generated ideal (Theorem) by pahio
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Cross-references: formally real, residue fields, rational function field, valuation rings, fractional ideal, generators, Prüfer rings, invertible ideals, Prüfer domains, principal ideal, proof, elements, generated by, Dedekind domain, ideal
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This is version 34 of two-generator property, born on 2004-02-27, modified 2007-07-30.
Object id is 5645, canonical name is TwoGeneratorProperty.
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AMS MSC11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)

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prufer domain, and two generator propert, unsolved? by nkadambi on 2006-03-11 07:52:55
Did not Schutling (1979) give an example of a Prufer domain that required 3 generators?

Schutling, Über die Erzeugendenanzahl invertierbarer Ideale in Pruferringen, Comm. Algebra, 7 (1979), no. 13, 1331-1349.
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