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 $\mathbb{T}_{n}(R)$ is left zero-clean. It is shown that a commutative ring $R$ is zero-clean if and only if the matrix ring $\mathbb{M}_{n}(R)$ is zero-clean for every positive integer $n\geq 1$. We characterize the zero-clean matrix rings over fields. We also determine when a $2\times2$ 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.