Nonsingular retractable modules and their endomorphism rings

Abstract

A module R M is said to be retractable if Hom R ( M, U ) ≠ 0 for each nonzero submodule U of M . M is said to be a CS module if every complement submodule of M is a direct summand in M . Retractable modules are compared to nondegenerate modules on the one hand and to e –retractable modules on the other (nondegenerate implies retractable implies e –retractable); and it is shown that if M is nonsingular and retractable, then End R M is a left CS ring if and only if M is a CS module.