Abstract view
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 $[K:k]
\leq 2$.