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'two-generator property'
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| Title of object: |
two-generator property |
| Canonical Name: |
TwoGeneratorProperty |
| Type: |
Corollary |
| Created on: |
2004-02-27 16:18:56 |
| Modified on: |
2004-02-28 10:17:10 |
| Classification: |
msc:11R04 |
| Synonyms: |
two-generator property=Ideal of Dedekind domain |
Preamble:
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Content:
Lemma (proved by Steinitz in 1911): Let $A$ and $B$ be ideals of a Dedekind domain $R$. Then there is an element $\omega$ and an ideal $C$ of $R$ such that
$$AC = (\omega)$$
$$B+C = R.$$
Theorem: Every ideal of a Dedekind domain can be generated by two of its elements.
Proof. Let $A$ be an arbitrary ideal of a Dedekind domain $R$. Let $B$ be such an ideal of $R$ that $AB$ is a principal ideal $(\alpha)$. The lemma gives also an element $\beta$ and an ideal $C$ of $R$ such that $AC = (\beta)$ and $B+C = R$. Then we have
$$A = g.c.d.(AB, AC) = g.c.d.((\alpha), (\beta)) = (\alpha, \beta)$$
because $g.c.d.(B, C) = B+C = R = (1)$.
Problem: Can the two-generator property be generalized to the invertible ideals of the Pr\"ufer domains (and the Pr\"ufer rings)? |
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