The quotients of the $p$-adic group ring of a cyclic group of order $p$

We classify, up to isomorphism, the ${\mathbb{Z}}_{p}G$-modules of rank $1$ (i.e., the quotients of ${\mathbb{Z}}_{p}G$) for $G$ cyclic of order $p$, where ${\mathbb{Z}}_p$ is the ring of $p$-adic integers. This allows us in particular to determine effectively the quotients of ${\mathbb{Z}}_{p}G$ which are cohomologically trivial over $G$. There are natural zeta functions associated to ${\mathbb{Z}}_{p}G$ for which we give an explicit formula.