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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.

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  • $\begingroup$ 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. $\endgroup$
    – Fernando Muro
    Commented Feb 21, 2020 at 11:26
  • $\begingroup$ yes that's what I meant by canonical, I will add this to the question, thanks $\endgroup$
    – nikola karabatic
    Commented Feb 21, 2020 at 11:28

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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 $.

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