Question

For any ring k, let A = Mn(k) and let R (resp. S) denote the ring of n × n upper (resp. lower) triangular matrices over k. Show that R is isomorphic to S.

Answer 1

To simplify the notations, we shall work in the (sufficiently typical)

case $n = 3$

Let $E =$

0 0 1

0 1 0

1 0 0

and let $\alpha$ be the inner automorphism of $A$ defined by $E$ (with $\alpha^2 = \operatorname{Id}_A$). An easy calculation shows that

$$\alpha \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right) = \left( \begin{array}{ccc} i & h & g \\ f & e & d \\ c & b & a \end{array} \right)$$

In particular, $\alpha$ restricts to a ring isomorphism from $R$ to $S$.