Epis and monos which must be isos
Orzech [1] has shown that every surjective endomorphism of a noetherian module is an isomorphism. Here we prove analogous results for injective endomorphisms of noetherian injective modules, and the duals of these results. We prove that every injective endomorphism, with large image, of a module wit...
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| Format: | Article |
| Language: | English |
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Wiley
1984-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171284000557 |
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| _version_ | 1849682982023987200 |
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| author | David J. Fieldhouse |
| author_facet | David J. Fieldhouse |
| author_sort | David J. Fieldhouse |
| collection | DOAJ |
| description | Orzech [1] has shown that every surjective endomorphism of a noetherian module is an isomorphism. Here we prove analogous results for injective endomorphisms of noetherian injective modules, and the duals of these results. We prove that every injective endomorphism, with large image, of a module with the descending chain condition on large submodules is an isomorphism, which dualizes a result of Varadarajan [2]. Finally we prove
the following result and its dual: if p is any radical then every surjective endomorphism of a module M, with kernel contained in pM, is an isomorphism, provided that every surjective endomorphism of pM is an isomorphism. |
| format | Article |
| id | doaj-art-d1e00fa8ad0841c5bc9d9a6aafb47f30 |
| institution | DOAJ |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1984-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-d1e00fa8ad0841c5bc9d9a6aafb47f302025-08-20T03:24:02ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251984-01-017350751210.1155/S0161171284000557Epis and monos which must be isosDavid J. Fieldhouse0Department of Mathematics and Statistics, University of Guelph, Guelph N1G 2W1, Ontario, CanadaOrzech [1] has shown that every surjective endomorphism of a noetherian module is an isomorphism. Here we prove analogous results for injective endomorphisms of noetherian injective modules, and the duals of these results. We prove that every injective endomorphism, with large image, of a module with the descending chain condition on large submodules is an isomorphism, which dualizes a result of Varadarajan [2]. Finally we prove the following result and its dual: if p is any radical then every surjective endomorphism of a module M, with kernel contained in pM, is an isomorphism, provided that every surjective endomorphism of pM is an isomorphism.http://dx.doi.org/10.1155/S0161171284000557injective endomorphismsurjective endomorphismascending chain condition (ACC)descending chain condition (DCC)artinian modulenoetherian moduleinjective moduleprojective moduleinjective hullprojective coversmall submodulelarge submodulepreradicalradicalidempotent preradical. |
| spellingShingle | David J. Fieldhouse Epis and monos which must be isos International Journal of Mathematics and Mathematical Sciences injective endomorphism surjective endomorphism ascending chain condition (ACC) descending chain condition (DCC) artinian module noetherian module injective module projective module injective hull projective cover small submodule large submodule preradical radical idempotent preradical. |
| title | Epis and monos which must be isos |
| title_full | Epis and monos which must be isos |
| title_fullStr | Epis and monos which must be isos |
| title_full_unstemmed | Epis and monos which must be isos |
| title_short | Epis and monos which must be isos |
| title_sort | epis and monos which must be isos |
| topic | injective endomorphism surjective endomorphism ascending chain condition (ACC) descending chain condition (DCC) artinian module noetherian module injective module projective module injective hull projective cover small submodule large submodule preradical radical idempotent preradical. |
| url | http://dx.doi.org/10.1155/S0161171284000557 |
| work_keys_str_mv | AT davidjfieldhouse episandmonoswhichmustbeisos |