If $f \in R[x]$ factors in $S[x]$ as $f(x) =g(x)h(x)$ with $g(x)$ a monic polynomial in $R[x]$, then $h$ is also in $R[x]$.
If $f$ divides $g$ in $S[x]$, show that $f$ divides $g$ in $R[x]$ for $R$ a sub-ring of $S$.