Fitriani, Fitriani and Wijayanti, Indah Emilia and Surodjo, Budi and Wahyuni, Sri and Faisol, Ahmad (2021) Category of Submodules of a Uniserial Module. Mathematics and Statistics, 9 (5). pp. 744-748. ISSN 2332-2071
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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 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 (HomR(N, K), HomR(N, L), HomR(N, M)) is a monic HomR(N, X)-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 (HomR(M, N), HomR(L, N), HomR(K, N)) is a monic HomR(X, N)-sub-exact, for all R-module N.
| Item Type: | Article |
|---|---|
| Subjects: | Q Science > QA Mathematics |
| Divisions: | Fakultas Matematika dan Ilmu Pengetahuan Alam (FMIPA) > Prodi Matematika |
| Depositing User: | AHMAD FAISOL |
| Date Deposited: | 11 Jul 2022 02:14 |
| Last Modified: | 11 Jul 2022 02:14 |
| URI: | http://repository.lppm.unila.ac.id/id/eprint/43161 |
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