Some Algebraic Classification of Semiregular Hypermodules in Connection to the Radical

We call a Krasner right S-hypermodule A regular if each cyclic subhypermodule of A is a direct summand of A, and we also call A semiregular if every finitely generated subhypermodule of A lies above a direct summand of A. In this study, some properties of such hypermodules are achieved. This paper consists of concentrating over the radical, as well as their connections with (semi)regularity of hypermodules. Finally, the relationship between f-supplement submodule and semiregularity is given.