Let $M$ be a $\Gamma$-ring and let $M_2=\begin{pmatrix} R&\Gamma\\ M &L\end{pmatrix}$, where $R$ (resp. $L$) is the right (resp. left) operator ring of $M$. In this paper we show that the class of all primitive subdirectly irreducible $\Gamma$-rings is special, and then we establish relationships between the antisimple primitive radicals of the $\Gamma$-ring $M$, the right operator ring $R$ of $M$, the matrix $\Gamma_{n,m}$-ring $M_{m,n}$, the $M$-ring $\Gamma$ and the ring $M_2$.