Relations between endomorphism rings, injectivity, surjectivity and uniserial modules

For a right module M-R over a ring R, we consider the set I of all the endomorphisms phi is an element of E := End(M-R) that are not injective and the set K of all the endomorphisms phi that are not surjective. We prove that. when M-R is a uniserial module, then E/K is a left chain domain and E/I is a right chain domain. The technique we make use of to prove this can be applied to arbitrary modules M-R, not-necessarily uniserial. When the endomorphisms phi are not in I (not in K), then left (right) divisibility in E corresponds to inclusion in the lattice L(M-R) of all submodules of M-R. This allows to study factorizations of injective (surjective, respectively) endomorphisms of M-R analyzing finite chains in the partially ordered set L(M-R).