Classes of Commutative Clean Rings

Iberkleid, Wolf; McGovern, Warren Wm.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. S1, pp. 101-110.

Soit $A$ une anneau commutatif unitaire et $I$ and idéal de $A$. L’anneau $A$ est dit $I$-propre si pour chaque élément $a\in A$ il existe un idempotent $e=e^2\in A$ tel que $a-e$ est une unité et que $ae\in I$. Un filtre $ℱ$ d’idéaux de $A$ est noetherien si pour tout $I\in ℱ$, il existe un idéal finiment engendré $J\in ℱ$ tel que $J\subseteq I$. Nous caractérisons les anneaux $I$-propres pour les idéaux $0$ $n\left(A\right)$, $J\left(A\right)$ et $A$ en termes du filtre multiplicatif noetherien des idéaux de $A$ ainsi que en termes de propriétés plus classiques de théorie des anneaux.

Let $A$ be a commutative ring with identity and $I$ an ideal of $A$. $A$ is said to be $I$-$clean$ if for every element $a\in A$ there is an idempotent $e=e^2\in A$ such that $a-e$ is a unit and $ae$ belongs to $I$. A filter of ideals, say $ℱ$, of $A$ is Noetherian if for each $I\in ℱ$ there is a finitely generated ideal $J\in ℱ$ such that $J\subseteq I$. We characterize $I$-clean rings for the ideals $0$, $n\left(A\right)$, $J\left(A\right)$, and $A$, in terms of the frame of multiplicative Noetherian filters of ideals of $A$, as well as in terms of more classical ring properties.

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