Integrality for PI-rings
type: Item
instance of: scholarly article
title: Integrality for PI-rings
zbMATH Open document ID: 0770.16011
DOI: 10.1016/0021-8693(92)90131-5
author: A. Braun
author: Nikolaus Vonessen
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: A ring \(S\) is called integral over a subring \(R\) if each element \(x\) of \(S\) satisfies an equation of the form \(x^ n+\sum p_ i(x)\), where the \(p_ i\) are words formed with \(x\) and elements of \(R\) with \(\text{deg}_ x(p_ i)<n\). This notion, introduced by \textit{W. Schelter} [ibid. 40, 245-257 (1976; Zbl 0341.16009)] for PI-rings, is thoroughly analyzed in the present paper. First, it is shown that lying over, going up, and incomparability hold for integral extensions of PI-rings, proving, among other things, that \(\text{Kdim}(R)= \text{Kdim}(S)\) if one of the two rings has classical Krull dimension \(\text{Kdim}\). Transitivity of integrality is another major result, generalizing earlier work by \textit{R. Paré} and \textit{W. Schelter} [ibid. 53, 477-479 (1978; Zbl 0404.16011)] for integral centralizing extensions, and by \textit{M. Artin} and \textit{W. Schelter} [Adv. Math. 39, 289-329 (1981; Zbl 0461.16014)] for affine PI-algebras over an algebraically closed field. A further topic are results of Eakin-Nagata type. Thus, it is shown that if \(R \subseteq S\) are PI-algebras over a commutative noetherian ring \(\Lambda\), such that \(S\) is left noetherian and affine over \(\Lambda\) and integral over \(R\), then \(_ RS\) is finitely generated and \(R\) is left noetherian and affine over \(\Lambda\). Perhaps the most important result is a powerful criterion for integrality, reducing the work to checking this for the center of \(S\): If \(S\) is prime, has its center integral over \(R\), and if \(R\) is noetherian modulo its nilradical \(N(R)\), then the trace ring \(T(S)\) is integral over \(R\). This allows several interesting applications to actions of finite groups and of Lie algebras over fields of prime characteristic, presented in the last section. Further topics include a discussion of integrality and embeddings of quotient rings, and an investigation of the relationship between integrality and module- finiteness. Thus, if \(R \subseteq S\…
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16U99
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 130648
zbMATH Keywords: PI-rings
zbMATH Keywords: lying over
zbMATH Keywords: going up
zbMATH Keywords: incomparability
zbMATH Keywords: integral extensions
zbMATH Keywords: classical Krull dimension
zbMATH Keywords: integrality
zbMATH Keywords: integral centralizing extensions
zbMATH Keywords: affine PI-algebras
zbMATH Keywords: center
zbMATH Keywords: trace ring
zbMATH Keywords: actions
zbMATH Keywords: embeddings of quotient rings
zbMATH Keywords: module-finiteness
zbMATH Keywords: PI-algebras
zbMATH Keywords: Schelter integrality
MaRDI profile type: Publication
full work available at URL: 0021-8693(92)90131-5
OpenAlex ID: W1987434726
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description: scientific article; zbMATH DE number 130648