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? 
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