Valued Modules over Skew Polynomial Rings 2

2016/05/04 by Gönenç Onay, Onay, Gönenç · 1 citation
Mathematics · #Advanced Topology and Set Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1605.01221

openalex publication_date 2016/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (\mathbbF_pⁿ(t)^alg)_{n∈ ℕ}, of fixed characteristic p>0, each equipped with the morphism x↦ x^pn and with the t-adic valuation.

Citations

• On a special class of polynomials

Cited by

• \mathbbF_p((X)) is decidable as a module over the ring of additive polynomials

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