Let $M$ be a left module over some ring $R$ and suppose that $M$ is an inverse limit of a family of modules $M_i$ with $i\in I$. We suppose also that the maps of the inverse limit $\pi_i:M\to M_i$ are all surjective.
Is it true that the composition length of $M$ is the supremum of the composition lengths of the $M_i$?
Note. By composition length I mean a function that takes values in $\mathbb N\cup \{\infty\}$, where $\infty$ is just a symbol, I do not want to distinguish among infinities.
Idea. My idea for a proof is to use the fact that the composition length of a module is just the length of the lattice of its submodules. Now, it is known that the length of a lattice which is the union of a directed family of sublattices is the supremum of the lengths of such sublattices. Now I am wandering if the dual lattice of the lattice of submodules of $M$ is the union of the directed family of the dual lattices of the submodules of the $M_i$. If this is true, one should conclude by showing that the length of a lattice equals the length of its dual... Do you think this can be arranged to work?
Remark. If the proof works like this, I think there will be no difficulty in arranging the whole thing in a Grothendieck category and not only in a category of modules. Am I right?