Let $R$ be a commutative ring. The first Weyl algebra over $R$ is the associative $R$-algebra generated by $x$ and $y$ subject to the relation $yx-xy=1$.
For which rings $R$, the first Weyl algebra $A_1(R)$ is simple?
If I am not wrong, it is necessary that $R$ will have characteristic zero. Is it true that for any intgral domain $D$ of characteristic zero, $A_1(D)$ is simple?
Theorem 2.19 is relevant for my question. It relies on Lemma 2.16.
Thank you very much.