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.