Minimal quasi-algebraic relations of skew derivations in characteristic \(p\geq 2\)
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 concepts like quasi-algebraic skew derivations, which are said to be left $Q$-algebraic modulo inner skew derivations. Examples of various results are also included.
scientific article; zbMATH DE number 7034175
scientific article