Semisimple modules that are small cyclic in their injective envelopes

This paper presents the fundamental characteristics of s-cosingular modules, which constitute semisimple and small submodules within an injective module. We establish that over a commutative Kasch ring S, each (semi) simple S-module is s-cosingular if and only if each maximal ideal of S is essential in S. Furthermore, we delve into the examination of modules that fulfill the condition of (S-s*). We provide several characterizations of rings using these modules. Specifically, we show that a ring S is left ss-Harada if and only if each left S-module verifies (S-s*).