On the endomorphism rings of max CS and min CS modules

The purpose of this study is to find a similar result of right-left symmetry of nonsingularity and max-min CS property on prime modules, in particular, on their endomorphism rings. The class of rings and modules with extending properties (i.e. CS, max CS, min CS, max-min CS) is an important class in ring and module theory. It attracts a lot of interest among ring theorists. Let R be an associative ring with identity and M, a right R− module. We prove that for a finitely generated, quasi-projective which is a self-generator M, it is a CS (resp. max CS, min CS, max-min CS) module if and only if its endomorphism ring S is right CS (resp. max CS, min CS, max-min CS). If M is a prime module, then M is nonsingular, max-min CS with a uniform submodule if and only if S is right and left nonsingular, right and left max-min CS with uniform right and left ideals. Moreover, if M is a semiprime, weak duo module, then M is max CS if and only if it is min CS.