What conditions make the ring of Laurent polynomials in non-commuting variables countable?

Question

Suppose we have some commutative ring $R$ and the ring of Laurent polynomials in a finite number of non-commuting variables $S=R\langle x_1,\,x_1^{-1},\ldots,\,x_n,\,x_n^{-1}\rangle$. Under what conditions on $R$ can we say $S$ is countable? Quite clearly this is the case when $R$ is countable, but under what conditions does this hold?

Answer 1

This is the case precisely when $R$ is itself countable (this follows by completely standard cardinality arguments).