Jacobson radicals of ring extensions.
type: Item
instance of: scholarly article
title: Jacobson radicals of ring extensions.
zbMATH Open document ID: 1267.16019
author: Jeffrey Bergen
author: Piotr Grzeszczuk
published in: Journal of Pure and Applied Algebra
publication date: 2013-03-06T00:00:00Z
review text: In this paper \(R\) is an algebra with multiplicative identity over a field \(K\). If \(\sigma\) is a \(K\)-linear automorphism of \(R\), then a \(\sigma\)-derivation \(d\) is a \(K\)-linear map \(d\colon R\to R\) such that \(d(rs)=d(r)s+\sigma(r)d(s)\) for all \(r,s\in R\). The ring \(R^d=\{r\in R:d(r)=0\}\) is called the ring of constants. A \(\sigma\)-derivation \(d\) is said to be locally nilpotent if for every \(r\in R\), there exists \(n=n(r)\geq 1\) such that \(d^n(r)=0\). If \(q\) is a nonzero element of \(K\), the authors call their \(\sigma\)-derivation \(q\)-skew if \(d\sigma(r)=q\sigma d(r)\) for all \(r\in R\). For subsets \(A,B\) of a ring \(R\), let \(\text{r.ann}_A(B)=\{a\in A:Ba=0\}\). \(R\) is said to satisfy the acc on right annihilators of powers if, for every \(r\in R\), there exists \(n\geq 1\) such that \(\text{r.ann}_R(r)\subseteq\text{r.ann}_R(r^2)\subseteq\cdots\subseteq\text{r.ann}_R(r^n)=\text{r.ann}_R(r^{n+1})=\cdots\). If \(0\neq r\in R\), \(r\) is called normalizing if \(rR=Rr\). An automorphism \(\sigma\) is said to have locally finite order if, for every \(r\in R\), there exists \(n\geq 1\) such that \(\sigma^n(r)=r\). The construction of the skew polynomial ring \(R[x;\delta]\) of automorphism type using a single derivation can be extended to construct the smash product \(R\#U(L)\), where \(L\) is a Lie algebra acting on \(R\) as derivations and \(U(L)\) is the universal enveloping algebra of \(L\). \(J(R[x;\delta])\) denotes the Jacobson radical of \(R[x;\delta]\). In the first part of this paper the authors extend existing results on the Jacobson radical of skew polynomial rings of derivation type when the base ring has no nonzero nil ideals. In particular, they show that if \(R\) is an algebra with no nonzero nil ideals satisfying the acc on right annihilators of powers, then \(J(R[x;\delta])=0\) if \(\delta\) is a derivation on \(R\). Moreover, \(J(R\#U(L))=0\) if \(L\) is a Lie algebra acting on \(R\) as derivations. They also …
reviewed by: Halina France-Jackson
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16S20
zbMATH DE Number: 6141847
zbMATH Keywords: Jacobson radical
zbMATH Keywords: skew polynomial rings of derivation type
zbMATH Keywords: rings of constants
zbMATH Keywords: skew derivations
zbMATH Keywords: locally nilpotent derivations
zbMATH Keywords: right annihilators of powers
zbMATH Keywords: normalizing elements
zbMATH Keywords: acc on right annihilators
MaRDI profile type: Publication
full work available at URL: j.jpaa.2012.03.023
OpenAlex ID: W2060166235
cites work: Radicals Of Polynomial Rings
cites work: Jacobson radical of skew polynomial rings and skew group rings
cites work: On Rings with Locally Nilpotent Skew Derivations
cites work: GOLDIE DIMENSION OF CONSTANTS OF LOCALLY NILPOTENT SKEW DERIVATIONS
cites work: Radicals of crossed products of enveloping algebras
cites work: Rings with Fixed-Point-Free Group Actions
cites work: Jacobson Radicals of Ore Extensions of Derivation Type
DOI: 10.1016/J.JPAA.2012.03.023
description: scientific article; zbMATH DE number 6141847