# Modules C-minimaux sur des anneaux de polyn\^omes tordus

## TL;DR

This paper characterizes $C$-minimal modules over non-commutative skew polynomial rings, showing certain valued modules, including Puiseux series and ultraproducts of valued fields, are $C$-minimal.

## Contribution

It provides a complete characterization of $C$-minimal valued modules over specific non-commutative skew polynomial rings, extending the understanding of $C$-minimality in valued modules.

## Key findings

- Puiseux series over finite fields are $C$-minimal as modules.
- Ultraproducts of algebraically closed valued fields with non-standart Frobenius are $C$-minimal.
- Characterization of $C$-minimal modules over non-commutative rings of skew polynomials.

## Abstract

In this article we study modules endowed with a ultrametric, from the point of view of the geometric notion $C$-minimality. We give a complete characterization of $C$-minimal valued modules over non-commutative rings of skew polynomials of the form $R:=K[t;\varphi]$, where $K$ is a field, $\varphi$ an endomorphism of $K$ and $R$ is the $K$-algebra generated by $t$, such that $at=ta^{\varphi}$ for $a\in K$. We deduce for instance that the ring of Puiseux series over a finite field $\mathbb{F}$ of characteristic $p>0$, as a valued module over $\mathbb{F}[t;x\mapsto x^p]$ is $C$-minimal. Moreover, any ultraproduct $\mathcal{K}$, of algebraically closed valued fields $\mathcal{K}_{p^n}$ of characteristic $p>0$, endowed each with the morphism $x\mapsto x^{p^n}$, following a ultrafilter $U$ over $\{p^n\ |\ n\in \mathbb{N}, \, \text{et} \; p \; \text{prime}\}$, equipped with the {\it non-standart Frobenius}, i.e., the map $\sigma_{U}:=\lim_{U} x \mapsto x^{p^n}$, is $C$-minimal as a $\mathcal{K}[t;\sigma]$-valued module.