| Summary: | The class of rings J = {A|(A, ◦) 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) V0
W0 J (S)
)
,
where J (R) and J (S) are the Jacobson radicals of R and S , respectively, V0 = {v ∈ V |vW ⊆ J (R)} and
W0 = {w ∈ W|wV ⊆ 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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