The quotients of the $p$-adic group ring of a cyclic group of order $p$
Maria Guedri, Yassine Guerboussa
TL;DR
This work classifies all rank-1 $\mathbb{Z}_pG$-modules for a cyclic group $G$ of order $p$, distinguishing finite and infinite cases and focusing on cohomologically trivial (CT) modules. The finite case yields a complete presentation: each $A$ fits into a short exact sequence with a unique submodule $M$ of $L=\mathbb{Z}_pG$, and $A$ is determined by invariants $r,s,t$ with $M$ explicitly described; CT occurs exactly when $t\neq 0$ for $rs\ge1$. The authors also provide explicit counts of all such quotients and CT quotients, and define associated zeta functions counting submodules and CT quotients. In the infinite case, the classification splits by torsion, giving concrete isomorphism types for torsion-free and torsion-containing modules, and establishing the corresponding decompositions and presentations. Collectively, the results give a precise, invariant-based description of rank-1 quotients of the $p$-adic group ring and connect the module theory to analytic objects via the zeta functions.
Abstract
We classify, up to isomorphism, the $\mathbb{Z}_pG$-modules of rank $1$ (i.e., the quotients of $\mathbb{Z}_pG$) 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 $\mathbb{Z}_pG$ which are cohomologically trivial over $G$. There are natural zeta functions associated to $\mathbb{Z}_pG$ for which we give an explicit formula.