Skew derivations whose invariants satisfy a polynomial identity
type: Item
instance of: scholarly article
title: Skew derivations whose invariants satisfy a polynomial identity
zbMATH Open document ID: 0957.16023
author: Jeffrey Bergen
author: Piotr Grzeszczuk
published in: Journal of Algebra
publication date: 2000-10-17T00:00:00Z
review text: Let \(R\) be an algebra over the field \(K\), \(\delta\) an additive map on \(R\), and \(\sigma\) an automorphism of \(R\). Call \(\delta\) a \(\sigma\)-derivation if \(\delta(xy)=\delta(x)y+\sigma(x)\delta(y)\) for \(x,y\in R\), and a \(q\)-skew \(\sigma\)-derivation if in addition \(\delta\sigma=q\sigma\delta\) for a nonzero \(q\in K\). The invariants of \(\delta\) are \(\{r\in R\mid\delta(r)=0\}=R^{(\delta)}\) and of \(\sigma\) are \(\{r\in R\mid\sigma(r)=r\}=R^{(\sigma)}\). Also, \(\delta\) is separable if \(R^{(\delta)}=\{r\in R\mid\delta^n(r)=0\), some \(n\geq 1\}\) and \(\sigma\) is separable if \(R^{(\sigma)}=\{r\in R\mid(\sigma-1)^n(r)=0\), some \(n\geq 1\}\). The authors investigate when the ring \(R^{(\delta)}\) satisfies a polynomial identity (PI) forces \(R\) to satisfy a PI also. The main result shows this to be the case when \(R\) is a prime algebra, \(\delta\) is \(q\)-skew \(\sigma\)-derivation, and both \(\delta\) and \(\sigma\) are algebraic over \(K\). One need only assume that \(\delta\) is a \(\sigma\)-derivation and that \(\delta\) is algebraic when \(R\) is a domain, or when \(\sigma\) is \(X\)-inner. When \(\delta\) is a separable \(q\)-skew \(\sigma\)-derivation of an arbitrary \(K\)-algebra \(R\), then \(\delta\) and \(\sigma\) algebraic force \(R\) to be a PI ring when \(R^{(\delta)}\) is, or when \(\sigma\) is separable and \(R^{(\delta)}\cap R^{(\sigma)}\) is PI.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16W22
zbMATH DE Number: 1471838
zbMATH Keywords: additive maps
zbMATH Keywords: automorphisms
zbMATH Keywords: skew derivations
zbMATH Keywords: invariants
zbMATH Keywords: polynomial identities
zbMATH Keywords: prime algebras
MaRDI profile type: Publication
full work available at URL: jabr.2000.8297
OpenAlex ID: W1989406677
cites work: ALGEBRAIC AUTOMORPHISMS AND $ PI$-ALGEBRAS
cites work: Actions of Commutative Hopf Algebras
cites work: Invariants of skew derivations
cites work: Invariants of \(U_q(\text{sl}(2))\) and \(q\)-skew derivations
cites work: Skew Derivations with Central Invariants
cites work: Q4865491
cites work: Group graded rings
cites work: Q4744461
cites work: Q4002121
cites work: Derivations et automorphismes algebriques d'Anneaux premiers
cites work: Q3786586
cites work: Algebras with a separable subalgebra whose centralizer satisfies a polynomial identity
cites work: Infinite Crossed Products and Group-Graded Rings
cites work: General polynomial identities. II
cites work: Rings with an integral element whose centralizer satisfies a polynomial identity
DOI: 10.1006/JABR.2000.8297
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description: scientific article; zbMATH DE number 1471838