A ring over which every finitely-generated left (or every finitely-generated right) module has a projective covering. A ring with Jacobson radical is a semi-perfect ring if and only if is semi-local and if every idempotent of the quotient ring has an idempotent pre-image in . The first condition can be replaced by the requirement of classical semi-simplicity of the quotient ring (cf. Classical semi-simple ring), and the second by the possibility of "lifting" modular direct decompositions from to . 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.
See also Perfect ring, and the references to that article.
Cf. also Projective covering.
|[a1]||L.H. Rowen, "Ring theory" , 1 , Acad. Press (1988) pp. 217|
Semi-perfect ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-perfect_ring&oldid=17322