I'm stumped with what should be an easy question from Stenström's Rings of Quotients: For a fixed module $V$, take $(\mathscr{T,F})$ to be the torsion theory cogenerated by $\{V\}$, and define the preradical $r_V$ as follows: $$r_V(M) = \bigcap\{\operatorname{ker}\varphi\mid \varphi \in \hom_R(M, V) \} $$
The question then is:
$(\mathscr{T,F})$ is hereditary if and only if $r_V = r_{E(V)}$
Where $E(V)$ is the injective hull of $E$.
We know that the torsion theory $(\mathscr{T},\mathscr{F})$ cogenerated by $V$, is the one that has as torsion-free class $\mathscr{F}$ the class of the modules cogenerated by $V$. That is $M\in \mathscr{F}$ if and only if there is mononomorphism $M\longrightarrow V^X$ for some set $X$. Now, $r_V$ is the idempotent radical asociated to this torsion theory. Is easy to see that $\mathscr{F}=\mathscr{F}_{r_V}$.
$\Rightarrow)$ We want to show that $\mathscr{F}_{r_V}=\mathscr{F}_{r_{E(V)}}$. To do this we use that these are the classes of modules cogenerated by $V$ and $E(V)$ respectively.
As $V\subseteq E(V)$, we have that $\mathscr{F}_{r_V}\subseteq\mathscr{F}_{r_{E(V)}}$.
We know that, $(\mathscr{T},\mathscr{F})$ is hereditary if and only if $\mathscr{F}$ is closed under injective envelopes. As $\mathscr{F}=\mathscr{F}_{r_V}$ is closed under injectivee envelopes and $V\in\mathscr{F}$, then $E(V)\in \mathscr{F}$. Therefore $\mathscr{F}_{r_V}=\mathscr{F}_{r_{E(V)}}$.
$\Leftarrow)$ As $r_V=r_{E(V)}$, we have that $\mathscr{F}_{r_V}\subseteq\mathscr{F}_{r_{E(V)}}$. Let $M\in\mathscr{F}$, then $M$ is submodule of $E(V)^X$ for some set $X$. It follows that $E(M)$ is submodule of $E(V)^X$. Thus $E(M)\in\mathscr{F}$. Therefore $(\mathscr{T},\mathscr{F})$ is hereditary.