On the existence of non-trivial finitely injective modules. (Q470228)

The main goal of the present paper is to characterize left Noetherian rings in terms of the existence of non-trivial finitely injective modules. Besides, the authors provide a simple way to construct non-trivial finitely injective modules.NEWLINENEWLINE Let \(R\) be a ring. Recall that a module \(M\) over \(R\) is said to be finitely injective if each finite subset of \(M\) is contained in an injective submodule of \(M\). Any direct sum of injective modules is trivially finitely injective. Thus, a finitely injective \(R\)-module \(M\) is said to be trivial if it is a direct sum of injective \(R\)-modules. This paper gives a positive answer to an open question of L. Salce by proving that a ring \(R\) is left Noetherian if and only if each finitely injective left \(R\)-module is trivial.