# When is Jacobson Radical a maximal ideal?

That in a ring that in a ring $$R$$ if the Jacobson radical is maximal ideal, then the ring $$R$$ is local?

I believe this is true because Jacobson Radical is intersection of maximal left ideals(hence it contains two sided maximal ideals) and hence the Jacobson Radical is contained in every maximal ideal. Because of maximality of Jacobson Radical, we have that any maximal ideal is $$J(R)$$ and thus $$R$$ has a unique maximal ideal. Hence $$R$$ is a local ring.

"The Jacobson radical of $$R$$ is a maximal ideal iff $$R$$ is local" is a true statement for commutative rings for the reasons you gave.
In noncommutative rings it is not: take for example $$M_2(\mathbb R)$$. The Jacobson radical is $$\{0\}$$, but there are uncountably many maximal left ideals.
At any rate it is true that if $$J(R)$$ is a maximal ideal, then it has to be the only maximal ideal. It is contained in (but not always equal to) the intersection of maximal two-sided ideals of $$R$$, since such ideals are also right primitive ideals.