Two Jacobson Radicals

Question

I have to know the Jacobson radicals of the rings $$ R=\left\{\left [\begin{array}\ \mathbb x & \mathbb y\\ 0 & \mathbb x \end{array} \right ] \mid x,y\in \mathbb Z_2\right\}$$ and $$S= \left\{\left [\begin{array}\ \mathbb x & \mathbb y\\ 0 & \mathbb x \end{array} \right ]\mid x\in \mathbb Z_4, y\in \mathbb Z_4 \oplus \mathbb Z_4\right\}.$$ I know that the Jacobson radical of $\left [\begin{array}\ \mathbb Z_2 & \mathbb Z_2\\ 0 & \mathbb Z_2 \end{array} \right ]$ is $\left [\begin{array}\ \mathbb 0 & \mathbb Z_2\\ 0 & \mathbb 0 \end{array} \right ]$, and this contains $J(R)$. Similarly, $J(S)$ is contained in $\left [\begin{array}\ 2\mathbb Z_4 & \mathbb Z_4\oplus\mathbb Z_4\\ 0 & 2\mathbb Z_4 \end{array} \right ]$. Any suggestion?

Answer 1

For both rings, if you take the quotient by the ideals you already found you obtain a ring of scalar matrices, thus a field. The intersectin of all the maximal subspaces is in both cases the zero ideal, so the inverse images of the zero ideal under the projections are the intersections of the maximal ideals, i.e. the ideals you already found.