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.
Asked
Active
Viewed 291 times
2
-
Are you want big list about it ? If not, see (https://math.stackexchange.com/questions/45085/an-example-of-a-division-ring-d-that-is-not-isomorphic-to-its-opposite-rin?rq=1) for a particular example! – Chinnapparaj R Apr 08 '19 at 07:20
-
The above link also contains a "biger list", namely this MO-post. James answers there:"Here's a factory for making examples." – Dietrich Burde Apr 08 '19 at 08:08
-
2@ChinnapparajR That example doesn't seem to help: a finite division ring is commutative, hence isomorphic to its opposite ring. But the link contained inside is helpful. – rschwieb Apr 08 '19 at 13:28
1 Answers
1
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.
rschwieb
- 160,592
-
1Adding this link in case I'm not the only reader who had never heard of Kasch rings. – Jyrki Lahtonen Apr 08 '19 at 13:46