Properties of Semi-Projective Modules and their Endomorphism Rings

Abstract

In this paper, we have studied the properties of semi-projective module and its endomorphism rings related with Hopfian, co-Hopfian, and directly finite modules. We have provide an example of module which are semi-projective but not quasi-projective. We also prove that for semi-projective module M with $dim M <\infty$ or $Codim M < \infty$, $M^n$ is Hopfian for every integer $n \ge 1$. Apart from this we have studied the properties of pseudo-semi-injective module and observed that for pseudo-semi-injective module, co-Hopficity weakly co-Hopficity and directly finiteness are equivalent. Finally proved that for pseudo-semi-injective module M, N be fully invariant M-cyclic submodule of M with N is essential in M, then N is weakly co-Hopfian if and only if M is weakly co-Hopfian.