Jacobson radical of upper triangular matrix rings

I suspect that the Jacobson radical of the ring

$$A = \begin{bmatrix} R & M \\ 0 & S \\ \end{bmatrix}$$

should be

$$\mathfrak{J}(A) = \begin{bmatrix} \mathfrak{J}(R) & M \\ 0 & \mathfrak{J}(S) \\ \end{bmatrix}$$

I know how to classify the left/right/two-sided ideals of this ring (T.Y. Lam, Chapter 1, Page 18) but I don't know how to find the maximal ones!

I'd like to see a proof of my conjecture if it's indeed true.

EDIT: I believe that this is not a duplicate of the linked question because first of all, this question is more general. Secondly, it requests the proof for something that is worth having its own separate question and thirdly, there's no proof in that linked question.

Solution 1:

You can confirm that if $m(R)$ is any maximal right ideal of $R$ and $m(S)$ is any maximal right ideal of $S$, then $\begin{bmatrix}m(R)&M\\0&S\end{bmatrix}$ and $\begin{bmatrix}R&M\\0&m(S)\end{bmatrix}$ are maximal right ideals of $A$.

The intersection of these as we range over the maximal right ideals of both rings is precisely $\begin{bmatrix}J(R)&M\\0&J(S)\end{bmatrix}$.