Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequences of divisible abelian groups. Does then the sequence splits?
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5$\begingroup$ This has nothing to do with algebraic geometry or linear algebra (although it may have applications in those areas), so it was not a good idea to add those tags. The commutative algebra tag that you started with was quite appropriate. Further, this looks more like a homework problem than it does a research problem, so it would probably be more suitable on MathStackExchange. $\endgroup$– Joe SilvermanCommented Mar 13, 2018 at 21:14
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2$\begingroup$ Since you used the ac tag, the keyword is "injective module". So, one half is the notion and basic properties of injective modules. The second half is that a divisible module over a PID is injective. Nothing very specific to abelian groups (i.e. modules over the specific ring $\mathbf{Z}$). $\endgroup$– YCorCommented Mar 13, 2018 at 22:37
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1 Answer
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Yes, because due to divisibility, $A$ is a direct summand in $B$, that is the embedding of $A$ into $B$ has a left inverse; this is the splitting lemma.
answered Mar 13, 2018 at 21:50
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$\begingroup$ Can you explain how can one reach a left inverse? this is exactly what is confusing me. $\endgroup$ Commented Mar 13, 2018 at 21:55
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1$\begingroup$ @user350168 see Theorem 21.2 in Fuchs' Infinite Abelian Groups. $\endgroup$ Commented Mar 13, 2018 at 21:59
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$\begingroup$ @TomaszKania could you tell me why does divisibility imply that $A$ is a direct summand in $B$? $\endgroup$– kuboCommented Apr 28 at 10:09
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