When is the radical associated with a module a torsion?

For an arbitrary $R$-module $M$ we consider the radical (in the sense of Maranda)$\mathfrak G_M, namely, the largest radical among all radicals $\mathfrak G$, such that$\mathfrak G(M)=0$. We determine necessary and sufficient on $M$ in order for the radical $\mathfrak G_M$ to be a~torsion. In particular,$\mathfrak G_M$ is a~torsion if and only if $M$ is a pseudo-injective module.