What are the submodules of the $\mathbb{Z}$-module $\mathbb{Z}[\frac{1}{p}]$? Given $p \in \mathbb{Z}$ a prime number and $M=\mathbb{Z}\left[\dfrac{1}{p}\right]$ a module over the ring $\mathbb{Z}$. How could I determine all submodules of M?
I thougt it would work to consider the generating set $\left\lbrace 1, \dfrac{1}{p}, \dfrac{1}{p^2},\ldots \right\rbrace$ and then each submodule would be generated by a modified subset (for example multiplicating each member of the subset with an integer) of it. But I am not sure about this.

$Z[1/P]$ is not like polynomial ring. otherwise, since $1/P$ is a root of the polynomial $Px-1=0$ of degree one, must $Z[1/P]=\{a: a \in Z\}=Z$ like $Z[\sqrt{2}]=\{a+b \sqrt{2}: a,b \in Z \}$ where $\sqrt{2}$ is a root of polynomial equation $X^2-2=0$ of degree two.
The proper submodules of $Z[1/P]$ are $Z[2/P]$, $Z[3/P]$, ..., $Z[(P-1)/P]$.
