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Correct classes of modules

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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Main Author: Wisbauer, Robert
Format: Article
Language:English
Published: Lugansk National Taras Shevchenko University 2018
Subjects:
Cantor-Bernstein Theorem
correct classes
homological classification of rings
16D70
16P40
16D60
Online Access:https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1014
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spelling oai:ojs.admjournal.luguniv.edu.ua: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
institution Algebra and Discrete Mathematics
collection 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
work_keys_str_mv AT wisbauerrobert correctclassesofmodules
first_indexed 2024-04-12T06:25:32Z
last_indexed 2024-04-12T06:25:32Z
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