Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local
cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp.
R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions
of local cohomology modules, and show that if the Lynch's conjecture, in , holds in
R[[X]], then it will holds in R.