Let $X$ be a module over some ring which splits as $$X\cong M\_1\oplus S\_1\cong M\_1\oplus M\_2 \oplus S\_2 \cong M\_1\oplus M\_2 \oplus M\_3\oplus S\_3\cong \ldots$$
where the isomorphisms come from splittings $S\_i\cong M\_{i+1}\oplus S\_{i+1}$.


Let $S=\bigcap S\_i$ and $M=\sum M\_i$. There is a canonical injection $M\oplus S\rightarrow X$ which is compatible with the projections to $S\_i$ and $M\_i$ in all of the above splittings.


**Question:** Is this an isomorphism, i. e. is $X\cong M\oplus S$? 


If the answer is "no", I would additionally be interested in some 'conceptual' way to 'measure' the failure of this being an isomorphism.



Limit of split short exact sequences
No. Take $X $ to be the direct product of nonzero modules $M\_i $ indexed by the positive integers, and $S\_i $ to be the direct product of all but the first $i $ of them. Then $S=0$ and $M $ is the direct sum of the $M\_i $.


