In D&F, they define split as below. (383pg of third edition)
Let $R$ be a ring and let $0 \rightarrow A \xrightarrow{\psi} B \xrightarrow{\phi} C \rightarrow 0$ be a short exact sequence of R-modules. The sequence is said to be split if there is an $R$-module complement to $\psi(A)$ in $B$. In this case, up to isomorphism, $B=A\oplus C$ (more precisely, $B=\psi(A)\oplus C'$ for some submodule $C'$, and $C'$ is mapped isomorphically onto $C$ by $\phi:\phi(C')\cong C$).
But then, isn't every short sequence is split? Since $\psi$ is injective and $\phi$ is surjective, $B/\psi(A) = B/Im\psi = B/ker\phi \cong Im\phi = C$, so that $B\cong \psi(A)\oplus C \cong A\oplus C$.
And also, they suggest an example(D&F, 384pg) of nonsplit exact sequence as $0 \rightarrow \mathbb{Z} \xrightarrow{n} \mathbb{Z} \xrightarrow{\pi} \mathbb{Z}/n\mathbb{Z} \rightarrow 0$ where $n$ sends $a$ to $na$ and $\pi$ is a natural projection. But, since $\mathbb{Z} \cong n\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}\cong \mathbb{Z}\oplus \mathbb{Z}/n\mathbb{Z}$, I can not find it is nonsplit.
There must be somewhere I misunderstood, but I can't find where. May I know which part is wrong?