A Jacobson radical of a matrix ring

Question

What is the Jacobson radical of the ring $R=\left [\begin{array}\ \mathbb Z_2 & \mathbb Z_2\\ 0 & \mathbb Z_4 \end{array} \right ]$? I think it is $\left [\begin{array}\ 0 & 0 \\ 0 & 2\mathbb Z_4 \end{array} \right ]$, but I am not sure. Could any body tells me how to obtain the radical? Thanks

Answer 1

Had you searched a bit, you would almost certainly have found one of these:

Jacobson radical of a certain ring of matrices

Jacobson radical of upper triangular matrix rings

Jacobson radical of a matrix ring

Jacobson radical of upper triangular matrix rings

They all carry sufficient explanation to explain why the radical of this ring is

\begin{bmatrix} 0&\mathbb Z_2 \\ 0& 2\mathbb Z_4 \end{bmatrix}