# Is there a non-commutative ring with unity $R$ such that its Jacobson radical is not a two-sided maximal ideal?

In the realm of commutative rings, it is well-understood that the Jacobson radical need not be a maximal ideal, as demonstrated by the ring $$R = \mathbb{Z}$$.

Does this principle also apply to non-commutative rings? Specifically, is there an example of a non-commutative ring with unity $$R$$ in which its Jacobson radical is not a two-sided maximal ideal?

Let $$k$$ be a field. The following matrix provides a counterexample:

$$\begin{pmatrix} k & k \\ 0 & k \end{pmatrix}$$

Are there other counterexamples available?

• It sounds a little bit like you're thinking of noncommutative rings and commutative rings as disjoint classes, whereas the prevailing viewpoint is that commutative rings are a subclass of "(possibly) noncommutative rings" Commented May 3 at 15:38

The Jacobson radical is not often maximal, because $$R/J(R)$$ does not have to be a simple ring.
Given any two nonzero rings $$R,S$$ at all, the Jacobson radical of $$R\times S$$ will not be maximal.
• DaRT query for rings that are "not top simple" meaning that $R/J(R)$ is not a simple ring. Commented May 3 at 15:37