Condensed Domains

Anderson, D. D. ; Dumitrescu, Tiberiu

Canadian mathematical bulletin, Tome 46 (2003) no. 1, pp. 3-13

Résumé

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 $[K:k]\,\le \,2$ .

Détail

DOI : 10.4153/CMB-2003-001-2

Mots-clés : 13A15, 13B22