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Extensions of clean rings. (English) Zbl 0989.16015

An element of a ring \(R\) is clean if it can be expressed as the sum of an idempotent and a unit in \(R\), and \(R\) is called a clean ring if every element of \(R\) is clean. The authors prove that if \(e\in R\) is an idempotent such that both \(eRe\) and \((1-e)R(1-e)\) are clean rings, then \(R\) is a clean ring. In particular, the matrix ring \(M_n(R)\) over a clean ring is clean. Other extensions of clean rings are studied, including group rings. The ring \(R[[x]]\) of formal power series over \(R\) is clean if and only if \(R\) is clean. However, if \(R\) is a commutative ring, the polynomial ring \(R[x]\) is not clean.

MSC:

16L30 Noncommutative local and semilocal rings, perfect rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16U60 Units, groups of units (associative rings and algebras)
16S50 Endomorphism rings; matrix rings
16S36 Ordinary and skew polynomial rings and semigroup rings
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
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References:

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