But what is the dual space of $Z_n$?
That would consist of all linear mappings from $Z_n$ into $Z$.
But there is only _one_ such mapping: the function that sends everything in $Z_n$ to $0$.
Suppose that we had linear $f$ such that $f(1) = k$.
Then $f(n * 1) = f(0) = 0$, and also $f(n * 1) = n * f(1) = n * k$.
So, in $Z$, we have $n * k = 0$, which means that $k = 0$ (given that $n >= 1$).
That implies that $f$ is the zero homomorphism.