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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?

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  • $\begingroup$ 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" $\endgroup$
    – rschwieb
    Commented May 3 at 15:38

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The Jacobson radical is not often maximal, because $R/J(R)$ does not have to be a simple ring.

It would seem you have answered your own question already, because your example is fine.

Given any two nonzero rings $R,S$ at all, the Jacobson radical of $R\times S$ will not be maximal.

Or, for example, any semiprimitive ring that is not simple.

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  • $\begingroup$ DaRT query for rings that are "not top simple" meaning that $R/J(R)$ is not a simple ring. $\endgroup$
    – rschwieb
    Commented May 3 at 15:37
  • $\begingroup$ Good idea! Thank you for sharing the information with me. $\endgroup$
    – Liang Chen
    Commented May 3 at 16:33

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