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Search: MSC category 13A15 ( Ideals; multiplicative ideal theory )

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1. CMB 2008 (vol 51 pp. 406)

Mimouni, Abdeslam
Condensed and Strongly Condensed Domains
This paper deals with the concepts of condensed and strongly condensed domains. By definition, an integral domain $R$ is condensed (resp. strongly condensed) if each pair of ideals $I$ and $J$ of $R$, $IJ=\{ab/a \in I, b \in J\}$ (resp. $IJ=aJ$ for some $a \in I$ or $IJ=Ib$ for some $b \in J$). More precisely, we investigate the ideal theory of condensed and strongly condensed domains in Noetherian-like settings, especially Mori and strong Mori domains and the transfer of these concepts to pullbacks.

Categories:13G05, 13A15, 13F05, 13E05

2. CMB 2003 (vol 46 pp. 3)

Anderson, D. D.; Dumitrescu, Tiberiu
Condensed Domains
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 $[K:k] \leq 2$.

Categories:13A15, 13B22

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