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Condensed Domains

Published online by Cambridge University Press:  20 November 2018

D. D. Anderson  and
Tiberiu Dumitrescu
D. D. Anderson
Affiliation:
Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419, U.S.A.
Tiberiu Dumitrescu
Affiliation:
Facultatea de Matematică, Universitatea Bucureşti, 14 Academiei Str., Bucharest 70109, Romania
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Abstract

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An integral domain $D$ with identity is condensed (resp., strongly condensed) if for each pair of ideals $I,\,J$ of $D,\,IJ\,=\,\{ij\,;\,i\,\in I,j\in J\,\}$ (resp., $IJ=iJ$ for some $i\,\in \,I$ or $IJ\,=Ij$ for some $j\,\in \,J$). We show that for a Noetherian domain $D,\,D$ is condensed if and only if $\text{Pic}\left( D \right)\,=0$ and $D$ is locally condensed, while a local domain is strongly condensed if and only if it has the two-generator property. An integrally closed domain $D$ is strongly condensed if and only if $D$ is a Bézout generalized Dedekind domain with at most one maximal ideal of height greater than one. We give a number of equivalencies for a local domain with finite integral closure to be strongly condensed. Finally, we show that for a field extension $k\,\subseteq K$, the domain $D=\,k+XK[[X]]$ is condensed if and only if $[K:k]\,\le \,2$ or $[K:k]\,=\,3$ and each degree-two polynomial in $k[X]$ splits over $k$, while $D$ is strongly condensed if and only if $[K:k]\,\le \,2$.

Keywords

13A1513B22
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 1 , 01 March 2003 , pp. 3 - 13
Copyright
Copyright © Canadian Mathematical Society 2003

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