Study of Semi-projective Retractable Modules

Abstract

For a semi-projective retractable module MR with endomorphism ring S, we prove u.dim MR= u.dim SS, and find necessary and sufficient conditions on M in order that S be respectively semiprime, right nonsingular, finitely cogenerated, cocyclic, or weakly co-Hopfian. Precise descriptions of the right singular ideal of S and the socle of M are given, and in addition if S is a semiprime ring, it is shown that MR is FI-extending if and only if SS is FI-extending.