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Since there are no comments to this question, I now restrict it to the following question:

It is known that: A ring $R$ is a prime left Goldie ring if and only if $R$ has a left quotient ring which is a matrix ring over a division ring (= simple Artinian); see, for example, Theorem 6.1.

Take $R=A_1(\mathbb{C})$, the first Weyl algebra over $\mathbb{C}$; it is a Noetherian domain, hence the above theorem can be applied, and we get that its left ring of quotients, denote it by $S$, is (isomorphic to) $M_n(D)$, $D$ a division algebra (infinite dimensional over $\mathbb{C}$).

Makar-Limanov (I could not download his article) mentions that $S$ is a division ring, so this means (if I am not wrong) that $n=1$.

(1) How to show that $n=1$? An available reference will help.

(2) From Artin-Wedderburn theorem, $D$ is the algebra of $S$-homomorphisms of a minimal nonzero left ideal of $S$; Can one describe $D$ more 'accurately'? (in terms of the generators $x$ and $y$ of $R$).

Thank you very much!

user237522
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    If $R$ is a domain then so is the left ring of quotients of $R$, but $M_n(D)$ is only a domain when $n=1$. This should answer (1) unless I am missing something? – stewbasic May 23 '17 at 22:22
  • You are right. I missed the claim that the left ring of quotients of a domain is a domain. Thanks for your comment! – user237522 May 23 '17 at 22:49
  • Here is a reference for the claim that a left Noetherian domain is a left Ore domain and the left quotient ring of a left Ore domain is a division ring:https://ysharifi.wordpress.com/2010/01/03/ore-domains-1-2/ – user237522 May 23 '17 at 23:02

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