Let $M$ be a $\mathrm{\Gamma }$-ring and let $M_2=\left(\begin{array}{cc}R& \mathrm{\Gamma }\\ M& L\end{array}\right)$, 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 $\mathrm{\Gamma }$-rings is special, and then we establish relationships between the antisimple primitive radicals of the $\mathrm{\Gamma }$-ring $M$, the right operator ring $R$ of $M$, the matrix ${\mathrm{\Gamma }}_{n,m}$-ring $M_{m,n}$, the $M$-ring $\mathrm{\Gamma }$ and the ring $M_2$.