# Confusion regarding the splitting lemma

My textbook, when discussing exact sequences of modules, makes the following claim.

If the short exact sequence $$0\to M'\xrightarrow{u} M\xrightarrow{v} M''\to 0$$ is exact and there is a morphism $$u':M'\to M$$ so that $$uu'=\text{id}_{M'}$$, it is easy to prove that $$M\cong\text{Ker}(u')\oplus\text{Im}(u)$$, and this means that this exact sequence splits.

However, the proof of this isn't obvious to me. Could anyone lend any hints? Many thanks, it's probably just a silly question.

• The composition $uu'$ doesn't make sense. Commented Mar 28, 2021 at 2:09
• Your $u’$ is wrong. Either it should go from $M$ to $M’$ and the composition you want is $u’u$; or else it should go from $M’’$ to $M$ and then you want the composition to be $u’v=\mathrm{Id}_{M’’}$. Also, you should state your definition of “the exact sequence splits” (there are several equivalent ways of defining it, so it’s important to know which one you are thinking about...) Commented Mar 28, 2021 at 2:11
• (Correction: you want $uu’$ even if $u\colon M\to M’$... too late to edit the comment) Commented Mar 28, 2021 at 2:18
• As a general principle with these sorts of questions, first ask yourself if there are any "obvious" maps between the things that are supposed to be isomorphic, in this case $M \to M' \oplus M''$ or the other way around. Then try to see if you can show that map is actually bijective. Commented Mar 28, 2021 at 3:54

Take the sequence $$0\rightarrow A \xrightarrow{\alpha} B \xrightarrow{\beta} C \rightarrow 0$$ an exact sequence.
And take $$u:B\rightarrow A$$ the inverse of $$\alpha$$, then the sequence splits. To prove this, you can take the morphism $$$$\psi: B \rightarrow A \oplus C\$$$$ defined by $$\psi(b)=u(b)+\beta (b)$$. Now it's easy to check that this is an isomorphism and this is the (correct) isomorphism
• It is incorrect to refer to $u$ as “the inverse of $\alpha$“. If $\alpha$ is not surjective, then it has no inverse. When the sequence splits, $\alpha$ has a left inverse. Commented Mar 28, 2021 at 17:41