two-generator propertyTwoGeneratorProperty2004-02-27 16:18:562007-07-30 03:52:48Theoremideals in a Dedekind domain% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}\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\, $\gcd(\mathfrak{b},\,\mathfrak{c}) = \mathfrak{b+c} = R = (1)$. $\Box$
The Dedekind domains are trivially Pr\"ufer domains, but the {\em two-generator property} can not be generalized to the invertible ideals of all Pr\"ufer domains (and Pr\"ufer rings):\, Sch\"ulting has constructed an invertible ideal of a Pr\"ufer domain that can not be generated by less than three generators.\, The example of Sch\"ulting is the fractional ideal\, $(1,\,X,\,Y)$\, of the Pr\"ufer 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.
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\bibitem{EM}{\sc Eben Matlis:} ``{The two-generator problem for ideals}''. \,-- {\em The Michigan Mathematical Journal} \textbf{17}\, $\mbox{N}\sp\circ$ 3 (1970).
\bibitem{HWS}{\sc Heinz-Werner Sch\"ulting:} ``{\"Uber die Erzeugendenanzahl invertierbarer Ideale in Pr\"uferringen}''. \,-- {\em Communications in Algebra} \textbf{7}\, $\mbox{N}\sp\circ$ 13 (1979). [Zentralblatt 432.13010]
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