Invertibility of Matrices over Subrings

Given rings $R\subseteq S$, consider the division closure $DC(R,S)$ and the rational closure $RC(R,S)$ of R in S. If S is commutative, then $DC(R,S)=RC(R,S)=R{T}^{-1}$, where $T=\{t\in R:t^{-1}\in S\}$. We show that this is also true if we assume only that R is commutative.