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• W2042863084 abstract "An element in a ring R is said to be clean (respectively unit-regular) if it is the sum (respectively product) of an idempotent element and an invertible element. If all elements in R are unit-regular, it is known that all elements in R are clean. In this note, we show that a single unit-regular element in a ring need not be clean. More generally, a criterion is given for a matrix (ab00) to be clean in a matrix ring M2(K) over any commutative ring K. For K=Z, this criterion shows, for instance, that the unit-regular matrix (12500) is not clean. Also, this turns out to be the “smallest” such example." @default.
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• W2042863084 date "2004-10-01" @default.
• W2042863084 modified "2023-10-12" @default.
• W2042863084 title "Clean matrices and unit-regular matrices" @default.
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• W2042863084 doi "https://doi.org/10.1016/j.jalgebra.2004.04.019" @default.
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