On injective and divisible modules. (Q1925589)

The author studies generalizations of injective and divisible modules. For a nonempty collection \(\mathfrak A\) of submodules of an \(R\)-module \(M\), the author defines an \(\mathfrak A\)-\(M\)-injective module and gets a characterization for a submodule \(M\) of a projective module \(P\) to be projective in terms of \(\{M\}\)-\(P\)-injective modules. In Section 2, the author studies \(n\)-injective (for a positive integer \(n\)) and \(f\)-injective modules. He gets some equivalent conditions for an \(n\)-injective module. In Section 3, the author studies \(n\)-divisible and \(f\)-divisible modules. He gets a characterization for \(n\)-divisibility in terms of \(\mathfrak C\)-\(F\)-injectivity for a free module \(F\) of rank \(n\) and the class of all cyclic submodules of \(F\). He gets a characterization of a PP-ring in terms of \(n\)-divisible and \(f\)-divisible modules. The author proves that for a PP-ring 1-divisibility implies \(f\)-divisibility.