Let $S$ be a local integral domain and $S[X]$ be a polynomial ring.

Choose $f, g$ from $S[X]$ as follows:

$f:= X^n + c_{n-1}X^{n-1} + ... + c_1X + c_0$

$g:= a_mX^m + ... + a_0$,

where $a_0, a_1,...,a_m$ all lie in the unique maximal ideal $m_S$ of $S$.

If $g$ is irreducible, does the ideal $(f,g)$ contain a non-trivial element of $S$ other than zero?