Factor Rings and their decompositions in the Eisenstein integers Ring ${\huge\mathbb{Z}}\left[ \omega \right]$

In this paper we will characterize the structure of factor rings for $\mathbb{Z}\left[ \omega \right]$ where $\omega=\frac{-1+\sqrt{-3}}{2},$is a 3rd primitive root of unity. Consequently, we can recognize prime numbers(elements) and their ramifications in $\mathbb{Z}\left[ \omega \right]$