Question

Compute the radical of the ring Tn(k) of n × n upper triangular matrices over any ring k.

Answer 1

To compute $\mathrm{rad}(\mathrm{T}_n(k))$ for any ring $k$ , we can treat $\mathrm{T}_n(k)$ as a triangular ring with $R = k$ , $S = \mathrm{T}_{n - 1}(k)$ , and $M = k^{n - 1}$ (as $(R, S)$ -bimodule). Using fact that $\mathrm{rad}(T) = \begin{pmatrix} \mathrm{rad}(R) & M \\ 0 & \mathrm{rad}(S) \end{pmatrix}$ and invoking an inductive hypothesis, we see that $\mathrm{rad}(\mathrm{T}_n(k))$ consists of $n \times n$ upper triangular matrices with diagonal entries from $\mathrm{rad}(k)$ .