Let $R$ be a commutative ring, and let $\mathfrak{a}\subseteq R$ be an ideal. For an $R$-module we consider the sub-$R$-modules $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{a}^n\subseteq(0:_Rx)\}$$ and $$\widetilde{\Gamma}_{\mathfrak{a}}(M)=\{x\in M\mid\mathfrak{a}\subseteq\sqrt{(0:_Rx)}\}$$ of $M$. These definitions give rise to left exact subfunctors $\Gamma_{\mathfrak{a}}$ and $\widetilde{\Gamma}_{\mathfrak{a}}$ of the identity functor on the category of $R$-modules, and $\Gamma_{\mathfrak{a}}$ is a subfunctor of $\widetilde{\Gamma}_{\mathfrak{a}}$.

Recall that a subfunctor $F$ of the identity functor is called a *radical* if $F(M/F(M))=0$ for every $R$-module $M$. Moreover, for every subfunctor $F$ of the identity functor there exists the smallest radical containing $F$.

Now, one can show that $\widetilde{\Gamma}_{\mathfrak{a}}$ is a radical, while $\Gamma_{\mathfrak{a}}$ need not be so. I conjecture but am unable to prove the following:

Conjecture:$\widetilde{\Gamma}_{\mathfrak{a}}$ is the smallest radical containing $\Gamma_{\mathfrak{a}}$.

Is anything known about this problem?

(*Motivation:* In the literature about torsion functors and their right derived functors (i.e., local cohomology) both definitions are used. Since most authors work over noetherian rings this does not matter: For an ideal of finite type, the two functors coincide. But if we wish to study non-noetherian situations it might be helpful to understand the precise relation between the two definitions.)