On the direct sum of dual-square-free modules
A module \(M\) is called square-free if it contains no non-zero isomorphic submodules \(A\) and \(B\) with \(A \cap B =0\). Dually, \(M\) is called dual-square-free if \(M\) has no proper submodules \(A\) and \(B\) with \(M =A +B\) and \(M/A \cong M/B\). In this paper we show that if \(M = \oplus _{...
Збережено в:
| Дата: | 2022 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2022
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1807 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| id |
oai:ojs.admjournal.luguniv.edu.ua:article-1807 |
|---|---|
| record_format |
ojs |
| spelling |
oai:ojs.admjournal.luguniv.edu.ua:article-18072022-06-15T04:49:44Z On the direct sum of dual-square-free modules Ibrahim, Y. Yousif, M. square-free and dual-square-free modules, abelian and quasi-duo rings Primary 16D40, 16D50, 16D60; Secondary 16L30, 16L60, 16P20, 16P40, 16P60 A module \(M\) is called square-free if it contains no non-zero isomorphic submodules \(A\) and \(B\) with \(A \cap B =0\). Dually, \(M\) is called dual-square-free if \(M\) has no proper submodules \(A\) and \(B\) with \(M =A +B\) and \(M/A \cong M/B\). In this paper we show that if \(M = \oplus _{i \in I}M_{i}\), then \(M\) is square-free iff each \(M_{i}\) is square-free and \(M_{j}\) and \( \oplus _{j \neq i \in I}M_{i}\) are orthogonal. Dually, if \(M = \oplus _{i =1}^{n}M_{i}\), then \(M\) is dual-square-free iff each \(M_{i}\) is dual-square-free, \(1 \leqslant i \leqslant n\), and \(M_{j}\) and \( \oplus _{i \neq j}^{n}M_{i}\) are factor-orthogonal. Moreover, in the infinite case, we show that if \(M = \oplus _{i \in I}S_{i}\) is a direct sum of non-isomorphic simple modules, then \(M\) is a dual-square-free. In particular, if \(M =A \oplus B\) where \(A\) is dual-square-free and \(B = \oplus _{i \in I}S_{i}\) is a direct sum of non-isomorphic simple modules, then \(M\) is dual-square-free iff \(A\) and \(B\) are factor-orthogonal; this extends an earlier result by the authors in [2, Proposition 2.8]. Lugansk National Taras Shevchenko University 2022-06-15 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1807 10.12958/adm1807 Algebra and Discrete Mathematics; Vol 33, No 1 (2022) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1807/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1807/865 Copyright (c) 2022 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
square-free and dual-square-free modules abelian and quasi-duo rings Primary 16D40 16D50 16D60; Secondary 16L30 16L60 16P20 16P40 16P60 |
| spellingShingle |
square-free and dual-square-free modules abelian and quasi-duo rings Primary 16D40 16D50 16D60; Secondary 16L30 16L60 16P20 16P40 16P60 Ibrahim, Y. Yousif, M. On the direct sum of dual-square-free modules |
| topic_facet |
square-free and dual-square-free modules abelian and quasi-duo rings Primary 16D40 16D50 16D60; Secondary 16L30 16L60 16P20 16P40 16P60 |
| format |
Article |
| author |
Ibrahim, Y. Yousif, M. |
| author_facet |
Ibrahim, Y. Yousif, M. |
| author_sort |
Ibrahim, Y. |
| title |
On the direct sum of dual-square-free modules |
| title_short |
On the direct sum of dual-square-free modules |
| title_full |
On the direct sum of dual-square-free modules |
| title_fullStr |
On the direct sum of dual-square-free modules |
| title_full_unstemmed |
On the direct sum of dual-square-free modules |
| title_sort |
on the direct sum of dual-square-free modules |
| description |
A module \(M\) is called square-free if it contains no non-zero isomorphic submodules \(A\) and \(B\) with \(A \cap B =0\). Dually, \(M\) is called dual-square-free if \(M\) has no proper submodules \(A\) and \(B\) with \(M =A +B\) and \(M/A \cong M/B\). In this paper we show that if \(M = \oplus _{i \in I}M_{i}\), then \(M\) is square-free iff each \(M_{i}\) is square-free and \(M_{j}\) and \( \oplus _{j \neq i \in I}M_{i}\) are orthogonal. Dually, if \(M = \oplus _{i =1}^{n}M_{i}\), then \(M\) is dual-square-free iff each \(M_{i}\) is dual-square-free, \(1 \leqslant i \leqslant n\), and \(M_{j}\) and \( \oplus _{i \neq j}^{n}M_{i}\) are factor-orthogonal. Moreover, in the infinite case, we show that if \(M = \oplus _{i \in I}S_{i}\) is a direct sum of non-isomorphic simple modules, then \(M\) is a dual-square-free. In particular, if \(M =A \oplus B\) where \(A\) is dual-square-free and \(B = \oplus _{i \in I}S_{i}\) is a direct sum of non-isomorphic simple modules, then \(M\) is dual-square-free iff \(A\) and \(B\) are factor-orthogonal; this extends an earlier result by the authors in [2, Proposition 2.8]. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2022 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1807 |
| work_keys_str_mv |
AT ibrahimy onthedirectsumofdualsquarefreemodules AT yousifm onthedirectsumofdualsquarefreemodules |
| first_indexed |
2024-04-12T06:26:06Z |
| last_indexed |
2024-04-12T06:26:06Z |
| _version_ |
1796109218235809792 |