# Limit of split short exact sequences

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.

• I guess you want more conditions, like the "canonical" injection being compatible with the projections onto the $M_i$ etc. With your current conditions it's easy to produce counterexamples. – Fernando Muro Feb 21 at 11:26
• yes that's what I meant by canonical, I will add this to the question, thanks – nikola karabatic Feb 21 at 11:28

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 $$M_i$$.