Category of Submodules of a Uniserial Module

Let R be a ring, K,M be R-modules, L a uniserial R-module, and X a submodule of L. The triple (K,L,M) is said to be X-sub-exact at L if the sequence K→X→M is exact. Let σ(K,L,M) is a set of all submodules Y of L such that (K,L,M) is Y -sub-exact. The sub-exact sequence is a generalization of an exact sequence. We collect all triple (K,L,M) such that (K,L,M) is an X-sub exact sequence, where X is a maximal element of σ(K,L,M). In a uniserial module, all submodules can be compared under inclusion. So, we can find the maximal element of σ(K,L,M). In this paper, we prove that the set σ(K,L,M) form a category, and we denoted it by CL. Furthermore, we prove that CY is a full subcategory of CL, for every submodule Y of L. Next, we show that if L is a uniserial module, then CL is a pre-additive category. Every morphism in CL has kernel under some conditions. Since a module factor of L is not a submodule of L, every morphism in a category CL does not have a cokernel. So, CL is not an abelian category. Moreover, we investigate a monic X-sub-exact and an epic X-sub-exact sequence. We prove that the triple (K,L,M) is a monic X-sub-exact if and only if the triple Z-modules ( , , ) is a monic -sub-exact sequence, for all R-modules N. Furthermore, the triple (K,L,M) is an epic X-sub-exact if and only if the triple Z-modules ( , , ) is a monic -sub-exact, for all R-module N.