Singly injective modules

Abstract

A left R-module M is said to be left singly injective if ExtR1(F/K,M) = 0 for any cyclic submodule K of any finitely generated free left R-module F. In this paper, we study the notion of singly injective modules which is generalization of injective modules and absolutely pure modules. In this direction, we give conditions which guarantee that each singly injective left R-module is either injective or absolutely pure. Finally, we study rings whose simple modules are singly injective (SSI-rings). © 2019 World Scientific Publishing Company.