On the restricted graded Jacobson radical of rings of Morita context

The class of rings J = {A vertical bar(A, degrees) forms a group} forms a radical class and it is called the Jacobson radical class. For any ring A, the Jacobson radical J (A) of A is defined as the largest ideal of A which belongs to 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 = (R V W S), we will show directly by the structure of the Jacobson radical of rings that the Jacobson radical J (T) = (J(R) V-0 W-0 J(S)), where J (R) and J (S) are the Jacobson radicals of R and S, respectively, V-0 = {v is an element of V vertical bar vW subset of J (R)} and W-0 = {w is an element of W vertical bar wV subset of 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.