Clean matrices and unit-regular matrices

By Dinesh Khurana, T. Lam. 2004

54 citations recorded by the source corpus.

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.

View Paper: https://www.semanticscholar.org/paper/0c24408a3960852112d25f92da02e292f5bdadbc

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Citations in the corpus

• Strongly clean triangular matrix rings over local rings (2007, result): “It should be noted that this definition is different from that given in [9].”