## Relative injectivity and CS-modules.(English)Zbl 0817.16004

Let $$R$$ be a ring with identity and let $$M$$ and $$N$$ be unital right $$R$$- modules. Let $$E(M)$$ denote the injective hull of $$M$$. A module is called CS if every submodule is essential in a direct summand. If $$\text{Hom}(N,E(M)) = 0$$ then $$M \oplus N$$ is CS if and only if $$M$$ and $$N$$ are CS and $$N$$ is $$M$$-injective. This generalizes a result of the author and B. J. Mueller [Osaka J. Math. 25, 531-538 (1988; Zbl 0715.13006)]. Moreover, if $$M$$ is nonsingular and $$N$$ is $$M$$-injective then $$M\oplus N$$ is CS if and only if $$M$$ and $$N$$ are both CS. It is also proved that any finite direct sum of relatively injective CS-modules is CS. The same result can be found in a paper of A. Harmanci and the reviewer [Houston J. Math. 19, 523-532 (1993; Zbl 0802.16006)].

### MSC:

 16D50 Injective modules, self-injective associative rings 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)

### Citations:

Zbl 0715.13006; Zbl 0802.16006
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