Prime radicals of constants of algebraic derivations.
type: Item
instance of: scholarly article
title: Prime radicals of constants of algebraic derivations.
zbMATH Open document ID: 1196.16036
author: Chen-Lian Chuang
author: Tsiu-Kwen Lee
published in: Monatshefte für Mathematik
publication date: 2010-04-30T00:00:00Z
full work available at URL: 26.pdf
review text: Let \(R\) be a prime algebra over a field \(F\), \(Q\) the symmetric quotient ring of \(R\), and \(C\) the extended centroid of \(R\). Any derivation of \(R\) extends to \(Q\), and any derivation \(\delta\) of \(Q\) is called a continuous derivation of \(R\) if \(\delta(I)\subseteq R\) for a nonzero ideal \(I\) of \(R\); \(\delta\) is an \(F\)-derivation if \(\delta(F)=0\). If \(\delta\) is an \(F\)-continuous derivation of \(R\) that is algebraic over \(F\), then it is well known that there is \(b\in Q\) so that \(\delta=ad(b)\) or \(\text{char\,}R=p\geq 2\) and \(g(\delta)=ad(b)\) for \(g(x)=h(x^p)\in F[x]\) with \(g(0)=0\); further, \(b\) is algebraic over \(F\) with minimal polynomial \(m(x)\in F[x]\). Finally, set \(R^{(\delta)}=\{x\in R\mid\delta(x)=0\}\) and for any ring \(S\) let \(P(S)\) be its prime radical. The main results in the paper relate the degree of nilpotence of \(P(R^{(\delta)})\) to the powers of the factors of \(m(x)\) in \(F[x]\). Specifically, if \(t\) is the largest power of any irreducible factor appearing in \(m(x)\) then \(P(R^{(\delta)})^{{2^t}-1}=0\), and for any overring \(S\) of \(P(R^{(\delta)})\) contained in \(C(b)=\{q\in Q\mid bq=qb\}\), \(P(R^{(\delta)})=R^{(\delta)}\cap P(S)\). The result as stated by the authors describes a potentially smaller exponent than \(t\). They also show that \(P(C(b))^{{2^t}-1}=0\).
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 5702943
zbMATH Keywords: derivations
zbMATH Keywords: prime rings
zbMATH Keywords: prime radical
MaRDI profile type: Publication
OpenAlex ID: W2153254762
Wikidata QID: Q57787927
cites work: Q4865491
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DOI: 10.1007/S00605-009-0104-4
description: scientific article; zbMATH DE number 5702943