# Classes of Commutative Clean Rings

Wolf Iberkleid; Warren Wm. McGovern

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

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

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topIberkleid, 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 -

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