The Jacobson Radical of a Matrix Algebra

Question

I am trying to solve the following question.

Let $A$ be the algebra over $\mathbb{C}$ consisting of matrices of the form \begin{pmatrix} * & * & 0 & 0 \\ * & * & 0 & 0 \\ * & * & * & 0 \\ * & * & * & * \\ \end{pmatrix} Find the Jacobson radical of $A$.

To be honest I don't really have any idea how to do this so any help would very much be appreciated!

Answer 1

We can use that post again here!

The construction there uses an $R-S$ bimodule to create the ring: $T=\begin{pmatrix} R & M \\ 0 & S \\ \end{pmatrix}$ with formal matrix multiplication as the product. We can use the same thing here, except we can use an $S-R$ bimodule in the lower left corner: $T=\begin{pmatrix} R & 0 \\ M & S \\ \end{pmatrix}$. In your case, $R=M=M_2(\mathbb{C})$ and $S$ is the subring of lower triangular matrices of $R$.

Applying (analogous) info from "that post", we have $rad(T)=\begin{pmatrix} rad(R) & 0 \\ M & rad(S) \\ \end{pmatrix}$.

Can you take it from here?

Extra tools to keep in mind when doing things like this:

• if $I$ is a nilpotent (one or two sided) ideal, then $I \subseteq rad(R)$.
• if $R/I$ is semisimple, then $rad(R)\subseteq I$.

Check and see what you get when you compute $T/rad(T)$ for your example! :)