Let $A$ be an AB3 abelian category. We will say that an object $M$ of $A$ is uniquely divisible if for any integer $n\neq 0$ the endomorphism $nid\_M$ is invertible. We will say that $M'$ is ind-torsion if it belongs to the smallest Serre subcategory of $A$ that contains all torsion objects (that is, those $N\in A$ for which there exists $n\neq 0$ such that $nid\_N=0$) and is closed with respect to $A$-coproducts.


My question is: which assumptions of $A$ are sufficient to ensure that there do not exist non-zero monomorphisms from a uniquely divisible $M$ into an ind-torsion $A'$? This statement is obvious for abelian groups, but I do not know how to generalize it further. 



When uniquely divisible objects can be embedded into ind-torsion ones?
If your category $A$ is AB5, the answer is positive, because the ind-torsion objects are only the objects $X$ of $A$ such that the canonical map from the colimit $\_\infty X$ of $\_n X$ to $X$ is an isomorphism, where $\_n X$ is the kernel of multiplication by $n$ on $X$ and the (filtered) colimit is taken over positive integers $n$ ordered by divisibility. (The only thing to check is stability under extensions of this class of objects; thanks to AB5 the proof is the same as for abelian groups.) If $M$ is such that $n.Id\_M$ is a mono for each integer $n>0$ (it is true for any subobject of a uniquely divisible object), then $M\to\, \_n X$ is always zero, so $M\to\,\_\infty X$ is always zero because $A$ is AB5.


I guess that AB4 is not enough.


