# Semi-perfect ring

A ring over which every finitely-generated left (or every finitely-generated right) module has a projective covering. A ring $R$ with Jacobson radical $J$ is a semi-perfect ring if and only if $R$ is semi-local and if every idempotent of the quotient ring $R/J$ has an idempotent pre-image in $R$. The first condition can be replaced by the requirement of classical semi-simplicity of the quotient ring $R/J$ (cf. Classical semi-simple ring), and the second by the possibility of "lifting" modular direct decompositions from $R/J$ to $R$. A semi-perfect ring may also be characterized by the condition that every module admits a direct decomposition with respect to which the maximal direct summands are complemented. A ring of matrices over a semi-perfect ring is a semi-perfect ring.