A generalization of a theorem of Faith and Menal and applications (Q1567228)

A ring \(R\) is a right V-ring if every simple right \(R\)-module is injective. A right \(R\)-module \(M\) is called a V-module if every simple right \(R\)-module is \(M\)-injective. \textit{C. Faith} and \textit{P. Menal} [Proc. Am. Math. Soc. 116, No. 1, 21-26 (1992; Zbl 0762.16011)] have established the V-ring theorem which gives a characterization of a V-ring. The author generalizes this theorem to V-modules and shows that, for example, \(M\) is a V-module if and only if there exists a semisimple module \(W\) satisfying \(I=r_Rl_W(I)\) for any right ideal \(I\) of \(R\) such that \(R/I\) is a submodule of \(M\)-generated modules. A ring \(R\) is a right Johns ring if \(R\) is right Noetherian and satisfies that any right ideal is a right annihilator ideal. If \(R\) is a right Johns ring, then \(R/J(R)\) is a V-ring by the V-ring theorem. A right Johns ring is a trivial Noetherian self-cogenerator. Using a right Johns ring and a strongly right Johns ring, the author constructs nontrivial modules which are Noetherian self-cogenerators.