On lifting and extending properties on direct sums of hollow uniform modules
A module \(M\) is said to be lifting if, for any submodule \(N\) of \(M\), there exists a direct summand \(X\) of \(M\) contained in \(N\) such that \(N/X\) is small in \(M/X\). A module \(M\) is said to satisfy the {\it finite internal exchange property} if, for any direct summand \(X\) of \(M\) an...
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| Дата: | 2022 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Lugansk National Taras Shevchenko University
2022
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1643 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| _version_ | 1856543296415858688 |
|---|---|
| author | Shibata, Y. |
| author_facet | Shibata, Y. |
| author_sort | Shibata, Y. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2022-06-15T04:49:44Z |
| description | A module \(M\) is said to be lifting if, for any submodule \(N\) of \(M\), there exists a direct summand \(X\) of \(M\) contained in \(N\) such that \(N/X\) is small in \(M/X\). A module \(M\) is said to satisfy the {\it finite internal exchange property} if, for any direct summand \(X\) of \(M\) and any finite direct sum decomposition \(M = \bigoplus_{i = 1}^n M_i\), there exists a direct summand \(M_i'\) of \(M_i\) \((i = 1, 2, \ldots, n)\) such that \(M = X \oplus (\bigoplus_{i = 1}^n M_i')\). In this paper, we first give characterizations for the square of a hollow and uniform module to be lifting (extending). In addition, we solve negatively the question ``Does any lifting module satisfy the finite internal exchange property?'' as an application of this result. |
| first_indexed | 2025-12-02T15:25:44Z |
| format | Article |
| id | admjournalluguniveduua-article-1643 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:25:44Z |
| publishDate | 2022 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-16432022-06-15T04:49:44Z On lifting and extending properties on direct sums of hollow uniform modules Shibata, Y. lifting modules, extending modules, finite internal exchange property 16D40, 16D70 A module \(M\) is said to be lifting if, for any submodule \(N\) of \(M\), there exists a direct summand \(X\) of \(M\) contained in \(N\) such that \(N/X\) is small in \(M/X\). A module \(M\) is said to satisfy the {\it finite internal exchange property} if, for any direct summand \(X\) of \(M\) and any finite direct sum decomposition \(M = \bigoplus_{i = 1}^n M_i\), there exists a direct summand \(M_i'\) of \(M_i\) \((i = 1, 2, \ldots, n)\) such that \(M = X \oplus (\bigoplus_{i = 1}^n M_i')\). In this paper, we first give characterizations for the square of a hollow and uniform module to be lifting (extending). In addition, we solve negatively the question ``Does any lifting module satisfy the finite internal exchange property?'' as an application of this result. 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/1643 10.12958/adm1643 Algebra and Discrete Mathematics; Vol 33, No 1 (2022) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1643/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1643/734 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1643/763 Copyright (c) 2022 Algebra and Discrete Mathematics |
| spellingShingle | lifting modules extending modules finite internal exchange property 16D40 16D70 Shibata, Y. On lifting and extending properties on direct sums of hollow uniform modules |
| title | On lifting and extending properties on direct sums of hollow uniform modules |
| title_full | On lifting and extending properties on direct sums of hollow uniform modules |
| title_fullStr | On lifting and extending properties on direct sums of hollow uniform modules |
| title_full_unstemmed | On lifting and extending properties on direct sums of hollow uniform modules |
| title_short | On lifting and extending properties on direct sums of hollow uniform modules |
| title_sort | on lifting and extending properties on direct sums of hollow uniform modules |
| topic | lifting modules extending modules finite internal exchange property 16D40 16D70 |
| topic_facet | lifting modules extending modules finite internal exchange property 16D40 16D70 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1643 |
| work_keys_str_mv | AT shibatay onliftingandextendingpropertiesondirectsumsofhollowuniformmodules |