Clean ring

Remarks

Example: a local ring $R$, one whose non-units form a two-sided ideal $\mathfrak{m}$. If $a$ is a unit, then $a = 0 + a$. If $a \in \mathfrak{m}$, then $a - 1 \notin \mathfrak{m}$, since otherwise $1 \in \mathfrak{m}$; so $a - 1$ is a unit and $a = 1 + (a - 1)$. Non-example: the ring $\Z$, whose idempotents are $0, 1$ and units are $\pm 1$. The sums $e + u$ are $-1, 0, 1, 2$, so $3$ is not clean. Every homomorphic image of a clean ring is clean, since idempotents and units map to idempotents and units.