Let D be an integral domain with quotient field K, M a torsion-free D-module, X an indeterminate, and $N_v={f{in}D[X]|c(f)_v=D}$. Let $q(M)=M{otimes}_D;K$ and $M_{w_D}$={$x{in}q(M)|xJ{subseteq}M$ for a nonzero finitely generated ideal J of D with $J_v$ = D}. In this paper, we show that $M_{w_D}=M[X]_{N_v}{cap}q(M)$ and $(M[X])_{w_{D[X]}}{cap}q(M)[X]=M_{w_D}[X]=M[X]_{N_v}{cap}q(M)[X]$. Using these results, we prove that M is a strong Mori D-module if and only if M[X] is a strong Mori D[X]-module if and only if $M[X]_{N_v}$ is a Noetherian $D[X]_{N_v}$-module. This is a generalization of the fact that D is a strong Mori domain if and only if D[X] is a strong Mori domain if and only if $D[X]_{N_v}$ is a Noetherian domain.