Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?


If not, is there an example?



Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?
This is true more generally for finitely generated modules over a noetherian ring. Your question is equivalent to asking whether the sequence 
$$0\rightarrow \operatorname{Hom}(B,A)\rightarrow \operatorname{Hom}(A\oplus B,A)\rightarrow \operatorname{Hom}(A,A)$$
is surjective on the right. To prove this, it suffices to localize and then complete at an arbitrary prime $P$, so we can assume we're working over a complete local ring where $P$ is the maximal ideal. Now it suffices to check surjectivity after modding out an arbitrary power $P^n$, which allows us to assume that all the modules are of finite length. Surjectivity follows because the lengths of the left-hand and right-hand modules add up to the length of the module in the middle.


**Edited to add:** The comments above (which I read after I posted this) remind me that I've posted this same argument before. If people think this instance should be deleted, I'm fine with that.


