Semisimple modules that are small cyclic in their injective envelopes

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