Correct classes of modules
For a ring \(R\), call a class \(\cal C\) of \(R\)-modules (pure-) mono-correct if for any \(M,N \in \cal C\) the existence of (pure) monomorphisms \(M\to N\) and \(N\to M\) implies \(M\simeq N\). Extending results and ideas of Rososhek from rings to modules, it is shown that, for an \(R\)-module...
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Lugansk National Taras Shevchenko University
2018
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admjournalluguniveduua-article-10142018-05-15T06:58:22Z Correct classes of modules Wisbauer, Robert Cantor-Bernstein Theorem, correct classes, homological classification of rings 16D70, 16P40, 16D60 For a ring \(R\), call a class \(\cal C\) of \(R\)-modules (pure-) mono-correct if for any \(M,N \in \cal C\) the existence of (pure) monomorphisms \(M\to N\) and \(N\to M\) implies \(M\simeq N\). Extending results and ideas of Rososhek from rings to modules, it is shown that, for an \(R\)-module \(M\), the class \(\sigma [M]\) of all \(M\)-subgenerated modules is mono-correct if and only if \(M\) is semisimple, and the class of all weakly \(M\)-injective modules is mono-correct if and only if \(M\) is locally noetherian. Applying this to the functor ring of \(R\)-Mod provides a new proof that \(R\) is left pure semisimple if and only if \(R\mbox{-Mod}\) is pure-mono-correct. Furthermore, the class of pure-injective \(R\)-modules is always pure-mono-correct, and it is mono-correct if and only if \(R\) is von Neumann regular. The dual notion epi-correctness is also considered and it is shown that a ring \(R\) is left perfect if and only if the class of all flat \(R\)-modules is epi-correct. At the end some open problems are stated. Lugansk National Taras Shevchenko University 2018-05-15 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1014 Algebra and Discrete Mathematics; Vol 3, No 4 (2004) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1014/543 Copyright (c) 2018 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
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2018-05-15T06:58:22Z |
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OJS |
| language |
English |
| topic |
Cantor-Bernstein Theorem correct classes homological classification of rings 16D70 16P40 16D60 |
| spellingShingle |
Cantor-Bernstein Theorem correct classes homological classification of rings 16D70 16P40 16D60 Wisbauer, Robert Correct classes of modules |
| topic_facet |
Cantor-Bernstein Theorem correct classes homological classification of rings 16D70 16P40 16D60 |
| format |
Article |
| author |
Wisbauer, Robert |
| author_facet |
Wisbauer, Robert |
| author_sort |
Wisbauer, Robert |
| title |
Correct classes of modules |
| title_short |
Correct classes of modules |
| title_full |
Correct classes of modules |
| title_fullStr |
Correct classes of modules |
| title_full_unstemmed |
Correct classes of modules |
| title_sort |
correct classes of modules |
| description |
For a ring \(R\), call a class \(\cal C\) of \(R\)-modules (pure-) mono-correct if for any \(M,N \in \cal C\) the existence of (pure) monomorphisms \(M\to N\) and \(N\to M\) implies \(M\simeq N\). Extending results and ideas of Rososhek from rings to modules, it is shown that, for an \(R\)-module \(M\), the class \(\sigma [M]\) of all \(M\)-subgenerated modules is mono-correct if and only if \(M\) is semisimple, and the class of all weakly \(M\)-injective modules is mono-correct if and only if \(M\) is locally noetherian. Applying this to the functor ring of \(R\)-Mod provides a new proof that \(R\) is left pure semisimple if and only if \(R\mbox{-Mod}\) is pure-mono-correct. Furthermore, the class of pure-injective \(R\)-modules is always pure-mono-correct, and it is mono-correct if and only if \(R\) is von Neumann regular. The dual notion epi-correctness is also considered and it is shown that a ring \(R\) is left perfect if and only if the class of all flat \(R\)-modules is epi-correct. At the end some open problems are stated. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1014 |
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AT wisbauerrobert correctclassesofmodules |
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2025-12-02T15:33:45Z |
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2025-12-02T15:33:45Z |
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1850411189258944512 |