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A note on tightness. (English) Zbl 0923.16003
For $$R$$-modules $$M$$ and $$N$$, $$M$$ is called weakly $$N$$-injective if for every homomorphism $$\phi\colon N\to E(M)$$, $$\phi(N)\subset X\simeq M$$ where $$X$$ is a submodule of the injective hull $$E(M)$$ of $$M$$. $$M$$ is called $$N$$-tight if every quotient of $$N$$ embeddable in $$E(M)$$ is embeddable in $$M$$. Every weakly $$N$$-injective module is $$N$$-tight. Generalizing earlier results of S. K. Jain and S. R. López-Permouth [J. Algebra 128, No. 1, 257-269 (1990; Zbl 0698.16012)] and S. K. Jain, S. R. López-Permouth and S. Singh [Glasg. J. Math. 34, No. 1, 75-81 (1992; Zbl 0747.16004)], the author proves that every cyclic right $$R$$-module is essentially embeddable in a projective module if $$R$$ is right Artinian and every indecomposable projective right module is uniform and $$R$$-tight.

##### MSC:
 16D50 Injective modules, self-injective associative rings
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