local cohomology and radical of ideal

Question

Let $R$ be commutative ring with identity, $M$ an $R$-module, and $I$ an ideal of $R$ . One defines $I$-torsion functor $Γ_I$ as: $\Gamma_I(M)=\bigcup_{n\in N} (0:_MI^n).$ When $R$ is Noetherian, it's known that $\Gamma_I(M)=\Gamma_{\sqrt I}(M)$.
What about non-Noetherian case? Is it true? or is there counterexample?

Thank you

Answer 1

Hint. $R=K[X_1,\dots,X_n,\dots]/(X_1^2,\dots,X_n^{n+1},\dots)$, $M=R$, and $I=(0)$.