Correct classes of modules

Correct classes of modules

For a ring R, call a class C of R-modules (pure-) mono-correct if for any M, N ∈ C the existence of (pure) monomorphisms M → N and N → M implies M ≃ N. Extending results and ideas of Rososhek from rings to modules, it is shown that, for an R-module M, the class σ[M] of all M-subgenerated modules...

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Published in:Algebra and Discrete Mathematics
Date:2004
Main Author: Wisbauer, R.
Format: Article
Language:English
Published: Інститут прикладної математики і механіки НАН України 2004
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/156603
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Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:Correct classes of modules / R. Wisbauer // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 106–118. — Бібліогр.: 18 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-156603
record_format dspace
spelling Wisbauer, R.
2019-06-18T17:44:23Z
2019-06-18T17:44:23Z
2004
Correct classes of modules / R. Wisbauer // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 106–118. — Бібліогр.: 18 назв. — англ.
1726-3255
2000 Mathematics Subject Classification: 16D70, 16P40, 16D60.
https://nasplib.isofts.kiev.ua/handle/123456789/156603
For a ring R, call a class C of R-modules (pure-) mono-correct if for any M, N ∈ C the existence of (pure) monomorphisms M → N and N → M implies M ≃ N. Extending results and ideas of Rososhek from rings to modules, it is shown that, for an R-module M, the class σ[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-Mod is pure-mono-correct. Furthermore, the class of pure-injective Rmodules 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.
en
Інститут прикладної математики і механіки НАН України
Algebra and Discrete Mathematics
Correct classes of modules
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Correct classes of modules
spellingShingle Correct classes of modules
Wisbauer, R.
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
author Wisbauer, R.
author_facet Wisbauer, R.
publishDate 2004
language English
container_title Algebra and Discrete Mathematics
publisher Інститут прикладної математики і механіки НАН України
format Article
description For a ring R, call a class C of R-modules (pure-) mono-correct if for any M, N ∈ C the existence of (pure) monomorphisms M → N and N → M implies M ≃ N. Extending results and ideas of Rososhek from rings to modules, it is shown that, for an R-module M, the class σ[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-Mod is pure-mono-correct. Furthermore, the class of pure-injective Rmodules 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.
issn 1726-3255
url https://nasplib.isofts.kiev.ua/handle/123456789/156603
citation_txt Correct classes of modules / R. Wisbauer // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 106–118. — Бібліогр.: 18 назв. — англ.
work_keys_str_mv AT wisbauerr correctclassesofmodules
first_indexed 2025-12-07T15:48:21Z
last_indexed 2025-12-07T15:48:21Z
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