Study of semi-projective retractable modules

For a semi-projective retractable module M-R with endomorphism ring S, we prove u.dim M-R = u.dim S-S, 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 M-R is FI-extending if and only if S-S is FI-extending.