تقرير
Noether's normalization in skew polynomial rings
| العنوان: | Noether's normalization in skew polynomial rings |
|---|---|
| المؤلفون: | Paran, Elad, Vo, Thieu N. |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Rings and Algebras |
| الوصف: | 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 $D[t_1,\ldots,t_n;\sigma_1,\ldots,\sigma_n]$ with respect to commuting automorphisms $\sigma_1,\ldots,\sigma_n$ of $D$. We give a sufficient condition for $\sigma_1,\ldots,\sigma_n$ 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. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2407.12686 |
| رقم الأكسشن: | edsarx.2407.12686 |
| قاعدة البيانات: | arXiv |
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