Strongly Simple-Injective Rings and Modules

If M and N are right R-modules, M is called simple-N-injective if every R-homomorphism γ from a submodule K of N into M with γ(K) simple extends to N. In this paper, a right R-module M is called strongly simple-injective if M is simple-N-injective for every right R-module N. We investigate the notion of strong simple-injectivity and, as an application, we provide a negative answer to a couple of questions on the subject, and a partial positive answer to another.