Bicommutators of fully divisible modules

We consider the module $M_R$, injective with respect to the pair $R/K\subseteq E(R/K)$, where $K=0:M$, and form the ring $Q_M (R)$ by constructing rings of quotients with respect to torsion. We find necessary and sufficient conditions on $M_R$ for $Q_M (R)$ to coincide with the bicommutator of $M_R$. Among the consequences of this are the well-known results of Beachy and Morita on bicommutators of coexact fully divisible modules and of injective endofinite modules.
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