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, S, R^op, S^op are all isomorphic.

Answer 1

One can also show that $R \sim R^{\mathrm{op}} \Rightarrow k \sim k^{\mathrm{op}}$ . Then since $R$ is isomorphic to $S$ (under mentioned assumptions), and $k$ has an anti-automorphism, and the same is true for $A, R$ and $S$ , then statement is obvious.

In details:

To simplify the notations, we shall work in the (sufficiently typical) case $n = 3$ . Suppose $\varepsilon : k \to k$ is an anti-automorphism (resp. involution). Composing the transpose map with $\varepsilon$ on matrix entries, we can define $\delta_0 : A \to A$ with

$$\delta_ {0} \left( \begin{array}{c c c} a & b & c \\ d & e & f \\ g & h & i \end{array} \right) = \left( \begin{array}{c c c} \varepsilon (a) & \varepsilon (d) & \varepsilon (g) \\ \varepsilon (b) & \varepsilon (e) & \varepsilon (h) \\ \varepsilon (c) & \varepsilon (f) & \varepsilon (i) \end{array} \right)$$

It is easy to check that this $\delta_0$ is an anti-automorphism (resp. involution) of $A$ , and therefore so is $\delta := \alpha \circ \delta_0$ given by

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

Thus $R, S, R^{\wedge}$ op, $S^{\wedge}$ op are all isomorphic.