Relative injectivity and CS-modules
In this paper we show that a direct decomposition of modules M⊕N, with N homologically independent to the injective hull of M, is a CS-module if and only if N is injective relative to M and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple mod...
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| Format: | Article |
| Language: | English |
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Wiley
1994-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171294000931 |
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| _version_ | 1849305205474066432 |
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| author | Mahmoud Ahmed Kamal |
| author_facet | Mahmoud Ahmed Kamal |
| author_sort | Mahmoud Ahmed Kamal |
| collection | DOAJ |
| description | In this paper we show that a direct decomposition of modules M⊕N, with N homologically independent to the injective hull of M, is a CS-module if and only if N is injective relative to M and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for quasi-injective modules. But when we confine ourselves to CS-modules we need no conditions on their socles. Then we investigate direct sums of CS-modules which are pairwise relatively inective. We show that every finite direct sum of such modules is a CS-module. This result is known for quasi-continuous modules. For the case of infinite direct sums, one has to add an extra condition. Finally, we briefly discuss modules in which every two direct summands are relatively inective. |
| format | Article |
| id | doaj-art-0235c8793e564c0b98f6f18f7598f4e5 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1994-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-0235c8793e564c0b98f6f18f7598f4e52025-08-20T03:55:32ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251994-01-0117466166610.1155/S0161171294000931Relative injectivity and CS-modulesMahmoud Ahmed Kamal0Ain Shams University, Faculty of Education, Mathematics Department, Heliopolis, Cairo, EgyptIn this paper we show that a direct decomposition of modules M⊕N, with N homologically independent to the injective hull of M, is a CS-module if and only if N is injective relative to M and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for quasi-injective modules. But when we confine ourselves to CS-modules we need no conditions on their socles. Then we investigate direct sums of CS-modules which are pairwise relatively inective. We show that every finite direct sum of such modules is a CS-module. This result is known for quasi-continuous modules. For the case of infinite direct sums, one has to add an extra condition. Finally, we briefly discuss modules in which every two direct summands are relatively inective.http://dx.doi.org/10.1155/S0161171294000931injective modulesself-injective ringsand generalization. |
| spellingShingle | Mahmoud Ahmed Kamal Relative injectivity and CS-modules International Journal of Mathematics and Mathematical Sciences injective modules self-injective rings and generalization. |
| title | Relative injectivity and CS-modules |
| title_full | Relative injectivity and CS-modules |
| title_fullStr | Relative injectivity and CS-modules |
| title_full_unstemmed | Relative injectivity and CS-modules |
| title_short | Relative injectivity and CS-modules |
| title_sort | relative injectivity and cs modules |
| topic | injective modules self-injective rings and generalization. |
| url | http://dx.doi.org/10.1155/S0161171294000931 |
| work_keys_str_mv | AT mahmoudahmedkamal relativeinjectivityandcsmodules |