Rings in which every element is the sum of a left zero-divisor and an idempotent

A ring R is called left zero clean if every element is the sum of a left zero-divisor and an idempotent. This class of rings is a natural generalization of O-rings and nil-clean rings. We determine when a skew polynomial ring is a left zero-clean ring. It is proved that a ring R is left zero-clean if and only if the upper triangular matrix ring T-n(R) is left zero-clean. It is shown that a commutative ring R is zero-clean if and only if the matrix ring M-n(R) is zero-clean for every positive integer n >= 1. We characterize the zero-clean matrix rings over fields. We also determine when a 2 x 2 matrix A over a field is left zero-clean. A ring is called uniquely left zero-clean if every element is uniquely the sum of a left zero-divisor and an idempotent. We completely determine when a ring is uniquely left zero-clean.