Inequality of module rank using annihilator

I am reading a paper and am somewhat confused by what I think should be a pretty elementary step, so any help would be really appreciated! Here's the question:

Let $R$ be a noetherian ring with noetherian ring of formal power series $R[[X]]$. Let $S \subseteq T$ be finitely-generated $R[[X]]$-modules and suppose that the quotient module $T/S$ is a finitely-generated $R$-module. If $T/S$ is annihilated by $X$, why does it follow that $\mathrm{rank}_{R} T/S \leq \mathrm{rank}_{R[[X]]} T$?

Thank in advance as always,

M