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...
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| Published in: | Algebra and Discrete Mathematics |
|---|---|
| Date: | 2004 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2004
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| 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| _version_ | 1862680941981335552 |
|---|---|
| author | Wisbauer, R. |
| author_facet | Wisbauer, R. |
| citation_txt | Correct classes of modules / R. Wisbauer // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 106–118. — Бібліогр.: 18 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| 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.
|
| first_indexed | 2025-12-07T15:48:21Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-156603 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T15:48:21Z |
| publishDate | 2004 |
| publisher | Інститут прикладної математики і механіки НАН України |
| 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 |
| spellingShingle | Correct classes of modules Wisbauer, R. |
| title | Correct classes of modules |
| title_full | Correct classes of modules |
| title_fullStr | Correct classes of modules |
| title_full_unstemmed | Correct classes of modules |
| title_short | Correct classes of modules |
| title_sort | correct classes of modules |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/156603 |
| work_keys_str_mv | AT wisbauerr correctclassesofmodules |