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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1994-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171294000931 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |