ring. Then R is semiperfect if any of the following equivalent conditions hold: R/J(R) is semisimple and idempotents lift modulo J(R), where J(R) is the Jacobson radical of R. R has a complete orthogonal set e1, ..., en of idempotents with each ei R ei a local ring. Every simple left (right) R-module has a projective cover. Every finitely generated left (right) R-module has a projective cover. The category of finitely generated projective -modules is Krull-Schmidt. Examples Examples of semiperfect rings include: Left (right) perfect rings. Local rings. Kaplansky's theorem on projective modules Left (right) Artinian rings. Finite dimensional k-algebras. Properties Since a ring R is semiperfect iff every simple left R-module has a projective cover, every ring Morita equivalent to a semiperfect ring is also semiperfect. Citations References Ring theory