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no. S1
Classes of Commutative Clean Rings
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 101-110.

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∈A there is an idempotent e=e 2 ∈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∈ℱ there is a finitely generated ideal J∈ℱ such that J⊆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.

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

@article{AFST_2010_6_19_S1_101_0,
     author = {Wolf Iberkleid and Warren Wm. McGovern},
     title = {Classes of {Commutative} {Clean} {Rings}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {101--110},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 19},
     number = {S1},
     year = {2010},
     doi = {10.5802/afst.1277},
     mrnumber = {2675723},
     zbl = {1210.13007},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1277/}
}
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Wolf Iberkleid; Warren Wm. McGovern. Classes of Commutative Clean Rings. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 101-110. doi : 10.5802/afst.1277. https://afst.centre-mersenne.org/articles/10.5802/afst.1277/

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