4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 Feb 21 at 11:26
  • $\begingroup$ yes that's what I meant by canonical, I will add this to the question, thanks $\endgroup$ – nikola karabatic Feb 21 at 11:28
8
$\begingroup$

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 $.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.