Turkish Journal of Mathematics




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

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