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.

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