I am trying to make sense of Definition 1.9 in the following paper. See my related question from earlier today.
Let $p$ be an odd prime and $Z_{(p)}$ be the localisation of $\mathbb{Z}$ at the prime ideal $p\mathbb{Z}$ i.e. $$Z_{(p)}=\lbrace\frac{a}{b}:a,b\in\mathbb{Z},b\notin p\mathbb{Z}\rbrace.$$ For each non-negative integer $k$, choose a generator $\alpha_k$ for the cyclic group of order $p^{v_p(k)+1}$. This cyclic group can then be written as $$\lbrace 1,\alpha_k,\alpha_k^2,\ldots,\alpha_k^{p^{v_p(k)}}\rbrace.$$ Looking at the first line defining the multiplications, it appears that there must be a $Z_{(p)}$-module structure on the above-defined cyclic group, but I can't see how this can be the case.
How can, say, $\frac{1}{2}\alpha_k$ be defined? It seems that we must be able to find fractional multiples of $\alpha_k$ for this definition to make any sense. Have I missed something?