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'two-generator property'
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| Title of object: |
two-generator property |
| Canonical Name: |
TwoGeneratorProperty |
| Type: |
Theorem |
| Created on: |
2004-02-27 16:18:56 |
| Modified on: |
2005-04-26 11:49:44 |
| Classification: |
msc:11R04 |
| Synonyms: |
two-generator property=Ideal of Dedekind domain |
Preamble:
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\newtheorem*{thmplain}{Theorem} |
Content:
\begin{thmplain}
\,\,Every ideal of a Dedekind domain can be generated by two of its elements.
\end{thmplain}
{\em 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 \,$\gcd(\mathfrak{b},\,\mathfrak{c}) = \mathfrak{b+c} = R = (1)$.
An open question is whether or not the two-generator property can be generalized to the invertible ideals of Pr\"ufer domains (and Pr\"ufer rings). |
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