Strongly injective modules

Abstract

Let R be a ring with identity and M be a left R-module. The module M is called strongly injective if whenever M+ K= N with M? N, there exists a submodule K? of K such that M?K?=N. In this paper, we provide the various properties of the class of these modules. In particular, we prove that M is strongly injective if and only if it is semisimple injective. Moreover, we give new characterizations of semisimple rings and left V-rings via strongly injective modules. Finally, we show that every strongly injective module is strongly noncosingular. © 2020, Springer-Verlag Italia S.r.l., part of Springer Nature.