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Semiclean rings and rings of continuous functions. (English) Zbl 1294.16025

Summary: As defined by Y. Ye [Commun. Algebra 31, No. 11, 5609-5625 (2003; Zbl 1043.16015)], a ring is semiclean if every element is the sum of a unit and a periodic element. M.-S. Ahn and D. D. Anderson [Rocky Mt. J. Math. 36, No. 3, 783-798 (2006; Zbl 1131.13301)] called a ring weakly clean if every element can be written as \(u+e\) or \(u-e\), where \(u\) is a unit and \(e\) an idempotent. A weakly clean ring is semiclean. We show the existence of semiclean rings that are not weakly clean. Every semiclean ring is 2-clean. New classes of semiclean subrings of \(\mathbf R\) and \(\mathbf C\) are introduced and conditions are given when these rings are clean. Cleanliness and related properties of \(C(X,A)\) are studied when \(A\) is a dense semiclean subring of \(\mathbf R\) or \(\mathbf C\).

MSC:

16U60 Units, groups of units (associative rings and algebras)
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16U80 Generalizations of commutativity (associative rings and algebras)
54C40 Algebraic properties of function spaces in general topology
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References:

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