Classes of Commutative Clean Rings

• Volume: 19, Issue: S1, page 101-110
• ISSN: 0240-2963

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Abstract

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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.

How to cite

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Iberkleid, Wolf, and McGovern, Warren Wm.. "Classes of Commutative Clean Rings." Annales de la faculté des sciences de Toulouse Mathématiques 19.S1 (2010): 101-110. <http://eudml.org/doc/115890>.

@article{Iberkleid2010,
abstract = {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 $\mathcal\{F\}$, of $A$ is Noetherian if for each $I \in \mathcal\{F\}$ there is a finitely generated ideal $J \in \mathcal\{F\}$ such that $J \subseteq I$. We characterize $I$-clean rings for the ideals $0$, $n(A)$, $J(A)$, 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.},
author = {Iberkleid, Wolf, McGovern, Warren Wm.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {clean rings; weakly clean ring},
language = {eng},
month = {4},
number = {S1},
pages = {101-110},
publisher = {Université Paul Sabatier, Toulouse},
title = {Classes of Commutative Clean Rings},
url = {http://eudml.org/doc/115890},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Iberkleid, Wolf
AU - McGovern, Warren Wm.
TI - Classes of Commutative Clean Rings
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2010/4//
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - S1
SP - 101
EP - 110
AB - 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 $\mathcal{F}$, of $A$ is Noetherian if for each $I \in \mathcal{F}$ there is a finitely generated ideal $J \in \mathcal{F}$ such that $J \subseteq I$. We characterize $I$-clean rings for the ideals $0$, $n(A)$, $J(A)$, 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.
LA - eng
KW - clean rings; weakly clean ring
UR - http://eudml.org/doc/115890
ER -

References

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6. Knox, M.L. and McGovern, W.W. Feebly projectable algebraic frames and multiplicative filters of ideals. Appl. Categ. Structures, to appear. Zbl1119.06008MR2306535
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8. Storrer, J.J. Epimorphismen von kommutativen Ringen. Comment. Math. Helvetici 43,378-401 (1968). Zbl0165.05301MR242810

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