Relations between endomorphism rings, injectivity, surjectivity and uniserial modules

For a right module MR over a ring R, we consider the set I of all the endomorphisms φ E:= End(MR) that are not injective and the set K of all the endomorphisms φ that are not surjective. We prove that when MR 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 MR, not-necessarily uniserial. When the endomorphisms φ are not in I (not in K), then left (right) divisibility in E corresponds to inclusion in the lattice (MR) of all submodules of MR. This allows to study factorizations of injective (surjective, respectively) endomorphisms of MR analyzing finite chains in the partially ordered set (MR).