Actions of solvable Lie algebras on rings with no nilpotent elements
type: Item
instance of: scholarly article
title: Actions of solvable Lie algebras on rings with no nilpotent elements
zbMATH Open document ID: 0707.16010
DOI: 10.1016/0021-8693(90)90159-L
author: Jeffrey Bergen
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let R be an algebra over a field K of characteristic \(p>0\) with no non- zero nilpotent elements, L be a finite dimensional solvable restricted Lie algebra over K and there is a Lie algebra homomorphism \(\phi: L\to Der_ K(R)\) which preserves pth powers. Let also \(R^ L=\{r\in R\); \(d(r)=0\), \(\delta\in \phi (L)\}\) be the ring of constants. The author proves the following main results: 1) if \(R^ L\) is a PI-algebra then R is also a PI-algebra; 2) R is left Goldie if and only if \(R^ L\) is left Goldie; 3) if R and \(R^ L\) are left Goldie then i) Q(R) is the localization of R at the regular elements of \(R^ L\); ii) \(Q(R^ L)=Q(R)^ L\) and \(R,R^ L\) have the same Goldie rank.
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16U20
zbMATH DE Number: 4160963
zbMATH Keywords: finite dimensional solvable restricted Lie algebra
zbMATH Keywords: ring of constants
zbMATH Keywords: PI-algebra
zbMATH Keywords: left Goldie
zbMATH Keywords: localization
zbMATH Keywords: Goldie rank
Wikidata QID: Q115365906
reviewed by: Yu. N. Maltsev
MaRDI profile type: Publication
cites work: Constants of Lie algebra actions
cites work: Actions of Commutative Hopf Algebras
cites work: Smash products and outer derivations
cites work: Q4073508
cites work: Derivations of prime rings
cites work: Rings with an integral element whose centralizer satisfies a polynomial identity
full work available at URL: 0021-8693(90)90159-l
OpenAlex ID: W2041206834
description: scientific article; zbMATH DE number 4160963