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On quasi-principally injective modules. (English) Zbl 0949.16003
A right $$R$$-module $$N$$ is said to be $$M$$-cyclic if it is isomorphic to $$M/L$$ for some submodule $$L\subseteq M$$, where $$M$$ is a right $$R$$-module. $$N$$ is defined to be $$M$$-principally injective if every homomorphism from an $$M$$-cyclic submodule of $$M$$ to $$N$$ can be extended to a homomorphism from $$M$$ to $$N$$. $$N$$ is called principally injective if it is $$R$$-principally injective. A module $$M$$ is called quasi-principally injective if it is $$M$$-principally injective and a ring $$R$$ is called right self-principally injective if $$R_R$$ is $$R$$-principally injective.
Characterizations of $$M$$-principally injective and quasi-principally injective modules are found, primarily in terms of some qualities of the endomorphism ring of $$M_R$$ and the right or left annihilators of certain ideals and elements.

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