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On weakly clean rings. (English) Zbl 1387.16035

Summary: A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by D. D. Anderson and V. P. Camillo [Commun. Algebra 30, No. 7, 3327–3336 (2002; Zbl 1083.13501)] and were extensively investigated by M.-S. Ahn and D. D. Anderson [Rocky Mt. J. Math. 36, No. 3, 783–798 (2006; Zbl 1131.13301)], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings.

MSC:

16U99 Conditions on elements
16S50 Endomorphism rings; matrix rings
16U60 Units, groups of units (associative rings and algebras)
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