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I'm trying to prove that a matrix ring of the form $\begin{pmatrix} R & R\\ 0 & Q \end{pmatrix}$ where $R/Q$ is an infinite degree field extension, is left Noetherian, but not right Noetherian. I was first looking at a concrete example, e.g. $\begin{pmatrix} \mathbb{R} & \mathbb{R}\\ 0 & \mathbb{Q} \end{pmatrix}$ which is also discussed here.

I think I have a solution for one part of the problem: If the extension $ R/Q$is of infinite degree, it means that there is an infinite sequence of field extensions $Q<K_1<K_2<…<R$. Looking at these $K_i$-s as $Q$-vector subspaces of $R$, using the construction given in the linked example, we get an infinite chain of right ideals, which means that it cannot be right Noetherian.

UPDATE: I think what I don't see right now is why such rings are left Noetherian. I think it would be convenient to tackle this using the fact that a module $M$ is (left) Noetherian iff for a submodule $N\leq M$, both $N$ and $M/N$ are (left) Noetharian.

sicmath
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    Well, what difficulty do you have generalizing from that example? – Eric Wofsey Mar 09 '23 at 22:51
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    To be fair, the solution there is only a hint in one direction. But still you should be able to use it to prove that same direction here... – rschwieb Mar 09 '23 at 22:53
  • There are also posts that explain how to classify all left and right ideals of such rings. – rschwieb Mar 09 '23 at 22:56
  • If $R/Q$ is of infinite degree, it means that there is an infinite sequence of field extensions $Q < K_1 < K_2 <\ldots < R$. Looking at these $K_i$-s as $Q$-vector subspaces of $R$, using the construction given in the linked example, we get an infinite chain of right ideals, which means that it cannot be right Noetherian. Is that correct? – sicmath Mar 09 '23 at 23:12
  • @sicmath Don't just put guesswork in a comment: take those thoughts and formalize them and put them into the post! – rschwieb Mar 10 '23 at 04:32
  • @sicmath I also think the idea in your update is worth pursuing. After all, it does reduce the problem to a question about two other modules which might be simpler. – rschwieb Mar 10 '23 at 04:33

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