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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.12686 |
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| _version_ | 1866909667613999104 |
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| author | Paran, Elad Vo, Thieu N. |
| author_facet | Paran, Elad Vo, Thieu N. |
| contents | 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 $R=F[t_1,\ldots,t_n]$ 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 $R=D[t_1,\ldots,t_n]$ 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 $D[t_1,\ldots,t_n;σ_1,\ldots,σ_n]$ with respect to commuting automorphisms $σ_1,\ldots,σ_n$ of $D$. We give a sufficient condition for $σ_1,\ldots,σ_n$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12686 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Noether's normalization in skew polynomial rings Paran, Elad Vo, Thieu N. Rings and Algebras 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 $R=F[t_1,\ldots,t_n]$ 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 $R=D[t_1,\ldots,t_n]$ 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 $D[t_1,\ldots,t_n;σ_1,\ldots,σ_n]$ with respect to commuting automorphisms $σ_1,\ldots,σ_n$ of $D$. We give a sufficient condition for $σ_1,\ldots,σ_n$ 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. |
| title | Noether's normalization in skew polynomial rings |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2407.12686 |