Determine if the following short exact sequence is split. Do the following short exact sequences split?
$$0\longrightarrow A\longrightarrow B\longrightarrow \mathbb{Z}^2 \longrightarrow 0$$
$$0\longrightarrow\mathbb{Z}\longrightarrow A\longrightarrow B\longrightarrow 0$$
This is a question on a Ph.D Topology exam.  I know what it means to be a split short exact sequence.  In order for the short exact sequence $0\longrightarrow A\longrightarrow B\longrightarrow C \longrightarrow 0$ split you need one of the following:


*

*there exists map $B\longrightarrow A$ such that $A\longrightarrow B\longrightarrow A$ is the identity on $A$. 

*there exists map $C\longrightarrow B$ such that $C\longrightarrow B\longrightarrow C$ is the identity on $C$.  

*$B$ is isomorphic to the direct sum of $A$ and $C$. 
I have tried to find examples and nonexamples online of split short exact sequences and I've tried to figure out how to answer the above question, but I am struggling hard.  I've tried to use the fact that these sequences are exact so we have the fact that the $Im(f_i)=Ker(f_{i+1})$. If someone could please give me an explanation to this question I would be very grateful. 
Thanks.

I assume the groups involved here are commutative. The first sequence splits because $Z^2$ is a free abelian group. if $f:B\rightarrow Z^2$, $f(u)=e_1,f(v)=e_2,$ where $e_1,e_2$ are generators of $Z^2$, write $g(e_1)=u, g(e_2)=v$.
