F-semi-perfect modules

Abstract

Let R be a ring with Jacobson radical J. Bass [ 1 ] called R semi-perfect if the factor ring R/J is semi-simple Artinian and every idempotent of R/J can be lifted to an idempotent of R. He showed that R is semi-perfect if and only if every cyclic left R-module has a projective cover and also that this so (if and) only if every finitely generated left R-module has a projective cover. The concept of semi-perfect rings has since then been generalized in two directions: to semi-perfect modules by Mares [3] and to F-semiperfect rings by Oberst and Schneider [6]. A left R-module is called a semi-perfect module if it is projective and every homomorphic image of it has a projective cover. Thus trivially R is a semi-perfect ring if and only if R is semi-perfect as a left R-module, and every finitely generated projective left module over a semi-perfect ring is semi-perfect. The following characterization was obtained: a projective left R-module P is semi-perfect if and only if JP is small in P, the factor module P/JP is completely reducible (=semi-simple), and every direct decomposition of P/JP can be lifted to that of P. On the other hand, a ring R is defined to be F-semi-perfect if the factor ring R/J is a regular ring (in the sence of von Neumann) and every idempotent of R/J can be lifted to an idempotent of R. It was proved that the F-semi-perfectness of R is equivalent to either of the following conditions: (a) every factor module R/Ra has a projective cover for a in R, or (b) every finitely presented left R-module has a projective cover. We attempt in this paper to generalize the concept of F-semi-perfect rings to modules along the line from semi-perfect rings to semi-perfect modules. Namely we call a left R-module P an F-semi-perfect module if B is projective and if, for every endomorphism s of P, the factor module P/Ps