Recall the following known result, Theorem 6.1: 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).
Now, let $R$ be an infinite dimensional $k$-algebra, which is a left Noetherian domain (a special case of prime left Goldie), $k$ is a field of characteristic zero. By the above result $R$ has a left quotient ring which is a matrix ring over a division ring, $M_n(D)$ ($D$ is infinite dimensional over $k$).
Is it possible to find $n$ and $D$? If this question is too general, then let us consider the special case $R=A_1(\mathbb{C})$, the first Weyl algebra over $\mathbb{C}$. Edit: I have found this paper by Makar-Limanov (I have not succeeded to download it), which mentions that the ring of quotients of the first Weyl algebra is a division ring, so in this case $n=1$.
Please see this relevant question.
Any hints and comments are welcome.
