# Valued Modules over Skew Polynomial Rings 2

## TL;DR

This paper extends the theory of ultrametic modules over skew polynomial rings, introducing new classes and proving quantifier elimination, with applications to valued difference fields and algebraically closed valued fields.

## Contribution

It introduces affinely maximal and residually divisible modules, proving quantifier elimination and Ax-Kochen–Ershov type theorems for valued modules over skew polynomial rings.

## Key findings

- Quantifier elimination for certain valued modules
- Axiomatization of ultraproducts of algebraically closed valued fields
- Application to valued difference fields with Frobenius endomorphism

## Abstract

Following our first article, we continue to investigate ultrametic modules over a ring of twisted polynomials of the form $[K;\vfi]$, where $\vfi$ is a ring endomorphism of $K$. The main motivation comes from the the theory of valued difference fields (including characteristic $p>0$ valued fields equipped with the Frobenius endomorphism). We introduce the class of modules, that we call, affinely maximal and residually divisible and we prove (relative -) quantifier elimination results. Ax-Kochen \& Erhov type theorems follows. As an application, we axiomatize, as a valued module, any ultraproduct of algebraically closed valued fields $(\mathbb{F}_{p^n}(t)^{alg})_{n\in \mathbb{N}}$, of fixed characteristic $p>0$, each equipped with the morphism $x\mapsto x^{p^n}$ and with the $t$-adic valuation.