I'm wondering if there is an infinite division ring $D$ and a finite unital subring $R$ of the full matrix ring $M_n(D)$ ($n$ some positive integer) such that there are no rings properly between $R$ and $M_n(D)$. Any ideas would be much appreciated! Thank you.
4
-
$\begingroup$ Could you say why this sort of question arises, and maybe either why you expect there is, or else that there isn't, such a datum? $\endgroup$– LSpiceCommented May 14, 2022 at 21:19
-
$\begingroup$ I see that a ring with a finite maximal subring must be finite, and so this answers the question. $\endgroup$– GregCommented May 15, 2022 at 5:30
-
$\begingroup$ @Greg If you know longer need an answer, can you either delete your question, or answer yourself and accept your answer? This way it won't clog up the "unanswered" queue. $\endgroup$– KimballCommented May 15, 2022 at 12:32
-
$\begingroup$ Oops - thanks for this and sorry for the delay. $\endgroup$– GregCommented May 19, 2022 at 1:26
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
If a ring has a finite maximal subring, the ring itself is finite. This appears to be due to Laffey from 1992.
-
$\begingroup$ Could you give a precise reference? $\endgroup$– Max HornCommented Oct 11, 2023 at 8:22