No. Take for $S$ the real power series in $t$ for which the coefficients of $t$ and $t^3$ both vanish. Take $f:=X^2-t^6$, $g:=t^5+t^2X$. We must check that there is no $h$ so that $gh$ is a nonzero constant modulo $f$. This is straightforward.

No. Take for $S$ the ring of real power series in $t$ for which the coefficients of $t$ and $t^3$ both vanish. Take $f:=X^2-t^6$, $g:=t^5+t^2X$. We must check that there is no $h$ so that $gh$ is a nonzero constant modulo $f$. This is straightforward.