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.


16D50 Injective modules, self-injective associative rings