Relative injectivity and CS-modules - Mahmoud Ahmed Kamal (International Journal of Mathematics and Mathematical Sciences, 1994)
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.