Semiperfect ring

Remarks

Example: a local ring $R$ is semiperfect: $R/J$ is a field, hence semisimple, and its only idempotents $0$ and $1$ lift to $0$ and $1$. Non-example: let $R = \set{a/b \in \Q : \gcd(b, 6) = 1}$. Its maximal ideals are generated by $2$ and $3$, so $R/J \cong \mathbb{F}_2 \times \mathbb{F}_3$ and clause 1 holds. Clause 2 fails: $R$ is a domain, so its only idempotents are $0$ and $1$, which cannot lift the idempotent $(1, 0)$ of $\mathbb{F}_2 \times \mathbb{F}_3$. Equivalent forms: every finitely generated left $R$-module has a projective cover; or $1 = e_1 + \cdots + e_n$ for orthogonal idempotents $e_i$ with each $e_i R e_i$ local. The definition is left-right symmetric.