The notion of generalized projective covers Is Introduced to give a natural generalization of a theorem of Bass on perfect rings. Moreover, In terms of this notion, some characterizations of semi-perfect rings and semi-perfect modules are established.

1. Semi-Perfect Rings Throughout R is a ring with identity element 1 whose Jacobson radical is denoted by J, and a semi-simple ring means an Artinian semi-simple ring, ie, a ring (with identity element) which is a (finite) direct sum of simple left ideals. Let M be a left R-module. Let P be a projective left R-module and ƒ: P→ M an epimorphism. P is called a projective cover of M with respect to ƒ if Ker (f) is small in P, ie, P= Ker (f)+ U with a submodule U of P always implies P= U. In his epoch-making paper [2], Bass investigated two important types of rings in connection with the concept of projective covers, one is" semi-perfect ring" and the other is" perfect ring". Namely, R is called semi-perfect if every cyclic left R-module has a projective cover, while R is called left (or right) perfect if every left (or right) R-module has a projective cover. The following were established by him, where the first one indicates that the concept of semi-perfect rings is left-right symmetric: