So I am thinking of a ring $R$ with multiplicative identity $1_{R}\ne 0_{R}$, $F$ is a free left $R$-module, and every element of $S$ is not a zero divisor, and $0_{R}\in S$. Under these conditions wouldn't it be trivial to conclude that $S^{-1}F$ is a free left $S^{-1}R$-module.
How would one prove such a statement?