Let $R$ be a nil-clean ring with unity such that $R/J(R)$ is reduced, where $J(R)$ is the Jacobson radical of $R$. Is it true that $R$ is abelian, i.e. the idempotents are central? (By nil-clean I mean each element is the sum of a nilpotent element and an idempotent element.)
By the first hypothesis, we could infer that each nilpotent element falls into $J(R)$. Also, by a nil-clean representation of $a-1$, where $a\in R$ is an arbitrary element, one deduces that each element is the sum of a unit and an idempotent.
Any help would be appreciated!