Studying semigroups using the properties of their prime $m$-ideals

In this article, we present the idea of $m$-ideals, prime $m$-ideals and their associated types for a positive integer $m$ in a semigroup. We present different chrarcterizations of semigroups through $m$-ideals. We demonstrate that the ordinary ideals, and their relevent types differ from the $m$-ideals and their assocated types by presenting concrete examples on the maximal, irreducible and strongly irreducible $m$-ideals. We conclude from the study that the introduction of the $m$-ideal will explore new fields of studies in semigroups and their applications.