On the restricted graded Jacobson radical of rings of Morita context

The class of rings $\mathcal{J}=\{A (A,\circ)$ forms a group$\}$ radical and it is called the Jacobson class. For any ring $A$, $\mathcal{J}(A)$ $A$ defined as largest ideal which belongs to $\mathcal{J}$. In fact, one most important classes since used widely in another branch abstract algebra, for example, construct two-sided brace. On other hand, every Morita context $T=\begin{pmatrix} R & V \\ W S \end{pmatrix}$, we will show directly by structure that $\mathcal{J}(T)= \begin{pmatrix} \mathcal{J}(R) V_0\\ W_0 \mathcal{J}(S) where $\mathcal{J}(R)$ $\mathcal{J}(S)$ are radicals $R$ $S$, respectively, $V_0=\{v \in vW \subseteq \mathcal{J}(R)\}$ $W_0=\{w wV \mathcal{J}(S)\}$. This clearly shows an $N-$radical. Furthermore, this property also valid restricted $G-$graded graded context.