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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| Date: | 2004 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2004
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| Series: | Algebra and Discrete Mathematics |
| Online Access: | http://dspace.nbuv.gov.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. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | 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. |
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