non-Dedekind Domain in which every ideal is generated by at most two elements - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:02:30Z http://mathoverflow.net/feeds/question/12969 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12969/non-dedekind-domain-in-which-every-ideal-is-generated-by-at-most-two-elements non-Dedekind Domain in which every ideal is generated by at most two elements Neal Harris 2010-01-25T21:40:37Z 2010-01-26T20:57:39Z <p>Does anyone know of such a domain?</p> http://mathoverflow.net/questions/12969/non-dedekind-domain-in-which-every-ideal-is-generated-by-at-most-two-elements/12972#12972 Answer by Ben Weiss for non-Dedekind Domain in which every ideal is generated by at most two elements Ben Weiss 2010-01-25T21:46:33Z 2010-01-26T07:37:03Z <p>[<b>Edited</b> to restrict to the case of quadratic orders. --PLC]</p> <p>Take any non-maximal order of a quadratic number field. This is not Dedekind because it fails to be integrally closed in its field of fractions. Every ideal is a free abelian subgroup of rank at most $2$. </p> <p>For example: $\mathbb{Z}[\sqrt{-3}].$</p> <p>I hope this answers your question. For further reading on Dedekind domains, and non-maximal Orders, I highly recommend the chapter on it in Neukirch's Algebraic Number Theory.</p> http://mathoverflow.net/questions/12969/non-dedekind-domain-in-which-every-ideal-is-generated-by-at-most-two-elements/12978#12978 Answer by Ben Linowitz for non-Dedekind Domain in which every ideal is generated by at most two elements Ben Linowitz 2010-01-25T22:10:32Z 2010-01-26T02:35:55Z <p>You may find Matlis' paper <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.mmj/1029000474" rel="nofollow">The Two-Generator Problem for Ideals</a> to be interesting, as its main theorem concerns the class of integral domains in which every ideal is generated by two elements.</p> <p>It was proven by Cohen (in <i>Commutative Rings with Restricted Minimum Condition</i> ) that an integral domain with the property that there exists an integer $n$ such that every ideal can be generated with fewer than $n$ elements must be Noetherian and of Krull dimension 1.</p> <p>Say that an integral domain R has property <b>FD</b> if every finitely generated torsion free R-module is direct sum of modules of rank 1. Moreover, say that R has property <b>FD locally</b> if R<sub>M</sub> has property FD for every maximal ideal M of R.</p> <p><b>Theorem</b> (simplified form) - Let R be an arbitrary integral domain. Then every ideal of R can be generated by two elements if and only if R is a noetherian ring that has property FD locally.</p> http://mathoverflow.net/questions/12969/non-dedekind-domain-in-which-every-ideal-is-generated-by-at-most-two-elements/13030#13030 Answer by Pete L. Clark for non-Dedekind Domain in which every ideal is generated by at most two elements Pete L. Clark 2010-01-26T10:15:40Z 2010-01-26T20:57:39Z <p>By way of comparison, Dedekind domains are characterized by an even stronger property, sometimes referred to colloquially as "$1+\epsilon$''-generation of ideals. Namely:</p> <p>Theorem: For an integral domain $R$, the following are equivalent:<br /> (i) $R$ is a Dedekind domain.<br /> (ii) For every nonzero ideal $I$ of $R$ and $0 \neq a \in I$, there exists $b \in I$ such that $I = \langle a,b \rangle$. </p> <p>The proof of (i) $\implies$ (ii) is such a standard exercise that maybe I shouldn't ruin it by giving the proof here. That (ii) $\implies$ (i) is not nearly as well known, although sufficiently faithful readers of Jacobon's <em>Basic Algebra</em> will know it: he gives the result as Exercise 3 in Volume II, Section 10.2 -- "Characterizations of Dedekind domains" -- and attributes it to H. Sah. (A MathSciNet search for such a person turned up nothing.) The argument is as follows: certainly the condition implies that $R$ is Noetherian, and a Noetherian domain is a Dedekind domain iff its localization at every maximal ideal is a DVR. The condition (ii) passes to ideals in the localization, and the killing blow is dealt by Nakayama's Lemma. </p> http://mathoverflow.net/questions/12969/non-dedekind-domain-in-which-every-ideal-is-generated-by-at-most-two-elements/13068#13068 Answer by Carl Weisman for non-Dedekind Domain in which every ideal is generated by at most two elements Carl Weisman 2010-01-26T20:28:23Z 2010-01-26T20:28:23Z <p>H. Sah was probably Chih Han Sah. Obituary: <a href="http://www.nytimes.com/1997/08/18/nyregion/chih-han-sah-62-mathematics-professor.html?pagewanted=1" rel="nofollow">http://www.nytimes.com/1997/08/18/nyregion/chih-han-sah-62-mathematics-professor.html?pagewanted=1</a></p>