Please introduce some different (non isomorphic) classes of finite rings which are not isomorphic to their opposite ring. I would like to study some examples.
2026-04-07 04:44:31.1775537071
FINITE Rings which are not isomorphic to their opposite
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The only one I have handy is a finite ring that is right Kasch but not left Kasch (so obviously it cannot be isomorphic to its opposite ring, which is left Kasch and not right Kasch.)
You can use any finite field $F$ and then look at the subring of matrices of $M_4(F)$ of the form
\begin{bmatrix} a&0&b&c\\ 0&a&0&d\\ 0&0&a&0\\ 0&0&0&e\end{bmatrix}
The DaRT entry is here, and the original source is given as T.-Y. Lam. Lectures on modules and rings. (2012) @ p 281.