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: | English |
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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 |
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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 |
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Algebra and Discrete Mathematics |
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| datestamp_date |
2022-06-15T04:49:44Z |
| collection |
OJS |
| language |
English |
| topic |
lifting modules extending modules finite internal exchange property 16D40 16D70 |
| 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 |
| topic_facet |
lifting modules extending modules finite internal exchange property 16D40 16D70 |
| format |
Article |
| author |
Shibata, Y. |
| author_facet |
Shibata, Y. |
| author_sort |
Shibata, Y. |
| title |
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_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_sort |
on lifting and extending properties on direct sums of hollow uniform modules |
| 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2022 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1643 |
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AT shibatay onliftingandextendingpropertiesondirectsumsofhollowuniformmodules |
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2025-12-02T15:25:44Z |
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2025-12-02T15:25:44Z |
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