On semi-artinian \(V\)-modules
The starting point of this paper is the fact that, as a consequence of a theorem of \textit{B. L. Osofsky} and \textit{P. F. Smith} [J. Algebra 139, 342-354 (1991; Zbl 0737.16001)], a right \(R\)-module \(M\) (over an arbitrary ring \(R\)) is semisimple if and only if every cyclic subquotient of \(M\) is \(M\)-injective. Observing that semisimple modules are semi-artinian (i.e., every nonzero image of \(M\) has nonzero socle) and are \(V\)-modules (i.e., every simple module is \(M\)-injective), the authors extend this fact in a significant way by proving that, if \(M\) is quasi-projective, then it is a semi-artinian \(V\)-module if and only if every nonzero cyclic subquotient of \(M\) contains a nonzero \(M\)-injective module. The paper ends by giving a method for producing examples of semi-artinian \(V\)-modules of prescribed socle length.
scientific article; zbMATH DE number 149932
scientific article