Mathematics and Statistics Vol. 9(5), pp. 744 - 748
DOI: 10.13189/ms.2021.090514
Reprint (PDF) (288Kb)

Category of Submodules of a Uniserial Module

Fitriani ^1,*, Indah Emilia Wijayanti ², Budi Surodjo ², Sri Wahyuni ², Ahmad Faisol ^1
1 Department of Mathematics, Universitas Lampung, Lampung, Indonesia
² Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia

ABSTRACT

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 C_L. Furthermore, we prove that C_Y is a full subcategory of C_L, for every submodule Y of L. Next, we show that if L is a uniserial module, then C_L is a pre-additive category. Every morphism in C_L has kernel under some conditions. Since a module factor of L is not a submodule of L, every morphism in a category C_L does not have a cokernel. So, C_L 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.

KEYWORDS
Sub-exact Sequences, Pre-additive Category, Uniserial Module

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Fitriani , Indah Emilia Wijayanti , Budi Surodjo , Sri Wahyuni , Ahmad Faisol , "Category of Submodules of a Uniserial Module," Mathematics and Statistics, Vol. 9, No. 5, pp. 744 - 748, 2021. DOI: 10.13189/ms.2021.090514.

(b). APA Format:
Fitriani , Indah Emilia Wijayanti , Budi Surodjo , Sri Wahyuni , Ahmad Faisol (2021). Category of Submodules of a Uniserial Module. Mathematics and Statistics, 9(5), 744 - 748. DOI: 10.13189/ms.2021.090514.