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International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 4, Pages 661-666

Relative injectivity and CS-modules

Ain Shams University, Faculty of Education, Mathematics Department, Heliopolis, Cairo, Egypt

Received 31 August 1992

Copyright © 1994 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this paper we show that a direct decomposition of modules MN, 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.