Let $R$ be a ring (assumed associative and unital) whose additive group is a finitely generated abelian group. As a reduction step in a paper I'm working on, we need to know that $R$ is a quotient of another ring $S$ whose additive group is a finite rank free abelian group. We believe we have a proof, but it is rather involved. Is this something that has appeared already or at least has a reasonably short proof? (The current proof reduces to showing it for finite rings and then proves it by using the basic ring associated to $R$.)

When $R$ is commutative, this has a quick proof: $R$ is generated by $r_1,\dots,r_n$ which are integral over ${\bf Z}$, and hence $R$ is a quotient of ${\bf Z}[x_1,\dots,x_n]/(f_1(x_1),\dots,f_n(x_n))$ where each $f_i$ is a monic univariate polynomial.

So I'm wondering if there is something analogous in the general case.