Minimal quasi-algebraic relations of skew derivations in characteristic \(p\geq 2\)
type: Item
instance of: scholarly article
title: Minimal quasi-algebraic relations of skew derivations in characteristic \(p\geq 2\)
zbMATH Open document ID: 1415.16038
author: Chen-Lian Chuang
published in: Journal of Algebra
publication date: 2019-03-11T00:00:00Z
review text: In this paper, the author supposes that $R$ is a prime ring with char $R =p\ge 2$ but some arguments are also valid for char $R=0$. In fact, the statements in the sequel should work for arbitrary characteristic, unless otherwise specified. Essentially, the author presents the monic minimal quasi-algebraic relation of an ordinary derivation $D$ of $R$ taking the form \[D^{ps}+ \sum^{s-1}_{i=0} \alpha_i D^{p^i}= \text{ad}(a),\] where $\alpha_i$ are in the extended centroid of $R$ and $a$ in the symmetric Martindale quotient ring of $R$. Among other things, he extends this result to an arbitrary skew derivation $D$. The author surveys in Section 0 related results in the literature, explains how this work is motivated and the current state. The article is organized as follows: In Section 1, the author sets up the context for working in $Q$. In Section 2, he introduces a generalization of $q$-skew derivations. The Section 3, the author defines $\delta^{ps}$ , for an $\sigma$-derivation $\delta$ such that $\sigma\delta\sigma^{-1}-\delta$ is $X$-inner. In addition, in the same section the author deduces the form of minimal quasi-algebraic relation of an $S$-derivation $D$ and pseudo $p^s$-powers of nearly 1-skew derivations. In Section 4, quasi-algebraic skew derivations and $cv$-polynomials, the author characterizes the monic minimal semi-invariant polynomial in terms of pseudo $p^i$th powers of the monic minimal cvpolynomials in $Q[X, S, D]$. By Lemma 2.3 and Proposition 2.4 of \textit{A. Leroy} and \textit{J. Matczuk} [J. Algebra 145, No. 1, 143--177 (1992; Zbl 0748.16011)], he deduces immediately a corresponding characterization of the monic minimal quasi-algebraic relation of $D$.\par The importance of the minimal quasi-algebraic relation of a skew derivation $D$ is that it basically determines all algebraic relations of $\{gD: g\in A\}$ or of $\{Dg: g\in A\}$, where $A$ form a group under composition. Additionally, the author mentions some relations between some conc…
reviewed by: Mehsin Atteya
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 7034175
zbMATH Keywords: prime ring
zbMATH Keywords: Martindale quotient ring
zbMATH Keywords: automorphism
zbMATH Keywords: skew derivation
zbMATH Keywords: differential identity
zbMATH Keywords: Ore extension
zbMATH Keywords: skew polynomial ring
zbMATH Keywords: semi-invariant polynomial
zbMATH Keywords: cv-polynomial
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2019.01.017
OpenAlex ID: W2911450869
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DOI: 10.1016/J.JALGEBRA.2019.01.017
description: scientific article; zbMATH DE number 7034175