Generators of $\displaystyle \mathbb{Z}$ are $\displaystyle <1> \ \text{and} \ <-1>$ correct?
$\displaystyle <n>$ denotes the subgroup generated by $\displaystyle n$.
So, $\displaystyle <1>=<-1>=\mathbb{Z}$
It doesn't make sense to say that the generators of $\displaystyle \mathbb{Z}$ are $\displaystyle \mathbb{Z}$ and $\displaystyle \mathbb{Z}$, which is what you essentially did.
See Example 1.4 of http://www.math.uconn.edu/~kconrad/b...ory/genset.pdf.