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Condensed Domains

  Published:2003-03-01
 Printed: Mar 2003
  • D. D. Anderson
  • Tiberiu Dumitrescu
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Abstract

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 $\Pic(D)=0$ 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 B\'{e}zout 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 $[K:k]\leq 2$ or $[K:k]=3$ and each degree-two polynomial in $k[X]$ splits over $k$, while $D$ is strongly condensed if and only if $[K:k] \leq 2$.
MSC Classifications: 13A15, 13B22 show english descriptions Ideals; multiplicative ideal theory
Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.)
13A15 - Ideals; multiplicative ideal theory
13B22 - Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.)
 

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