Primitive algebras with arbitrary Gelfand-Kirillov dimension
type: Item
instance of: scholarly article
title: Primitive algebras with arbitrary Gelfand-Kirillov dimension
zbMATH Open document ID: 0926.16024
author: Uzi Vishne
published in: Journal of Algebra
publication date: 1999-09-29T00:00:00Z
review text: Primitive algebras of arbitrary Gelfand-Kirillov dimension \(\geq 2\) were first constructed by \textit{R. S. Irving} and \textit{R.~B.~Warfield}, jun. [J. Lond. Math. Soc., II. Ser. 36, 219-228 (1987; Zbl 0638.16016)]. However, their examples required the base field \(k\) to have an infinite dimensional algebraic extension. In the paper under review, the author removes this restriction on \(k\). His examples are generalizations of the Morse algebra; they are monomial algebras, in fact, quotients of the free algebra \(k\langle x,y\rangle\) by ideals generated by carefully chosen sequences of monomials. The primitive algebras that arise have some very interesting additional features: their only two-sided ideals are the zero ideal together with the powers of the Jacobson radical; so although the Gelfand-Kirillov dimension of these examples can be any real number \(\geq 2\), their classical Krull dimension is always equal to one. Furthermore, all homomorphic images modulo nonzero ideals are finite dimensional, but the algebras are neither left nor right Noetherian.
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 1251299
zbMATH Keywords: growth
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: affine primitive algebras
zbMATH Keywords: Morse algebras
zbMATH Keywords: monomial algebras
zbMATH Keywords: Jacobson radical
zbMATH Keywords: classical Krull dimension
MaRDI profile type: Publication
OpenAlex ID: W2158052150
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DOI: 10.1006/JABR.1998.7567
description: scientific article; zbMATH DE number 1251299