General Linear Group $GL_n(R)$ not Finitely Generated

Question

Let $R$ an infinite field (or more general an infinite ring).

Let consider for $n \ge 2$ the general linear group $GL_n(R)$ and the special linear group $SL_n(R)$ considered with canonical matrix multiplication as group operation.

How to see that $GL_n(R)$ is not finitely generated as group?

Answer 1

Consider determinants. If $F$ is an infinite field, the determinants of a generating set of $\textrm{GL}_n(F)$ must generate the multiplicative group $F^*$. But $F^*$ contains a copy of either $\Bbb Q^*$, $\Bbb F_p(T)^*$ or is torsion but infinite, so cannot be finitely generated.

If $R$ is an infinite ring, then $\textrm{GL}_n(R)$ can be finitely generated, for instance when $R=\Bbb Z$.

Answer 2

They are uncountable and every finitely generated group is either finite or countable.