I want to find the Jacobson radical $J(R)$ of upper triangular block matrices
$$R=\begin{pmatrix}
A & B\\
0 & C
\end{pmatrix}$$
where $A\in\text{Mat}(n × n, D)$, $B\in \text{Mat}(n × k, D)$, $C \in \text{Mat}(k × k, D)$, $D$ is a division ring.
$J(R)$ is the intersection of all maximal left ideals and I have proved that the left ideals of $R$ are of the form
\begin{pmatrix}
I & B\\
0 & K
\end{pmatrix}
where $I$, $K$ are left ideals of $A$, $C$.
How should I find maximal ones from matrices of this form?
Best Answer
Firstly, your characterization of left ideals is incorrect. For example in the ring of $2\times 2$ upper triangular matrices, the elements of the form $\begin{bmatrix}x&x \\ 0& 0\end{bmatrix}$ form a left ideal, and are not of the form you describe.
But it is true that the maximal left ideals look like $\begin{bmatrix}L&B \\ 0& C\end{bmatrix}$ when $L$ is a maximal left ideal of $A$ and $\begin{bmatrix}A&B \\ 0& L\end{bmatrix}$ when $L$ is a maximal left ideal of $C$.
That is enough to show the Jacobson radical is $\begin{bmatrix}J(A)&B \\ 0& J(C)\end{bmatrix}$ . Furthermore, you know the Jacobson radical of square matrix rings over a division ring is zero, so...