It is well known that $\mathbf{PGL}_2(\mathbb{Z})$ is finitely generated, and that $\mathbf{PGL}_2(\mathbb{Q})$ isn't. My question is: what is a fast, natural way to see these properties without explicit construction of generators  ?

For instance, do (some) generating properties of an integral domain (or division ring) $D$ carry over to $\mathbf{PGL}_2(D)$  ?

It is well known that $\mathbf{PGL}_2(\mathbb{Z})$ is finitely generated, and that $\mathbf{PGL}_2(\mathbb{Q})$ isn't. My question is: what is a fast, natural way to see these properties without explicit construction of generators?

For instance, do (some) generating properties of an integral domain (or division ring) $D$ carry over to $\mathbf{PGL}_2(D)$?