Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Let $I$ an ideal of $R$. We have $$0:_MI = \cap_x(0:_Mx),$$ where $x$ runs a set of generators of $I$.
Now set $S = R[T]$ with $T$ is a variable. We have $M\otimes S = M[T]$ and $IS = I[T]$ and $$0:_{M[T]}IS = (0:_MI)[T]$$ by the flat extension property.
Question: Does there exist an element $X \in IS$ such that $$0:_{M[T]}IS = (0:_{M[T]}X)?$$
In my problem we can assume that $M$ is also Artinian (so has finite length).