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:

1. there exists map $B\longrightarrow A$ such that $A\longrightarrow B\longrightarrow A$ is the identity on $A$.
2. there exists map $C\longrightarrow B$ such that $C\longrightarrow B\longrightarrow C$ is the identity on $C$.
3. $B$ 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.