Noether's normalization in skew polynomial rings

2024/07/17 by Elad Paran, Paran, Elad, Thieu N. Vo +1 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2407.12686

openalex publication_date 2024/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study Noether's normalization lemma for finitely generated algebras over a division algebra. In its classical form, the lemma states that if I is a proper ideal of the ring R=F[t₁,…,t_n] of polynomials over a field F, then the quotient ring R/I is a finite extension of a polynomial ring over F. We prove that the lemma holds when R=D[t₁,…,t_n] is the ring of polynomials in n central variables over a division algebra D. We provide examples demonstrating that Noether's normalization may fail for the skew polynomial ring D[t₁,…,t_n;σ₁,…,σ_n] with respect to commuting automorphisms σ₁,…,σ_n of D. We give a sufficient condition for σ₁,…,σ_n under which the normalization lemma holds for such ring. In the case where D=F is a field, this sufficient condition is proved to be necessary.

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• Noether's normalization in iterated skew polynomial rings

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