Clean matrices and unit-regular matrices

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 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 is not clean. Also, this turns out to be the “smallest” such example.