Gelfand-Kirillov dimension of Noetherian semigroup algebras
type: Item
instance of: scholarly article
title: Gelfand-Kirillov dimension of Noetherian semigroup algebras
zbMATH Open document ID: 0801.16029
published in: Journal of Algebra
publication date: 1994-12-01T00:00:00Z
review text: Let \(K\) be a field, let \(S\) be a semigroup, and let \(K[S]\) be a semigroup algebra of \(S\) over \(K\). The author proves some theorems concerning the Gelfand-Kirillov dimension of \(K[S]\). His approach is based on his own results about the same problem for \(S\) cancellative, and on Grigorchuk's theorem describing semigroups with polynomial growth. The main tool used by the author is the connection between \(K[S]\) and \(K[\overline{S}]\), where \(\overline{S}\) is the image of \(S\) under the natural homomorphism \(K[S] \to K[S] / B(K[S])\) (\(B(R)\) is the prime radical of a ring \(R\)). The following proposition (Theorem 16) shows the character of the author's results. Denote by \(\text{GK}(S)\) the Gelfand-Kirillov dimension of \(K[S]\), and assume that \(K[S]\) is right Noetherian. Then the following conditions (i)-(iii) are equivalent: (i) \(\text{GK}(S) < \infty\), (ii) \(\text{GK}(T) < \infty\) for every cancellative subsemigroup \(T\) of \(S\), (iii) every cancellative subsemigroup of \(S\) has a finitely generated nilpotent-by-finite group of fractions. Moreover, in this case \(\text{GK}(S) \leq rq\), where \(r\) denotes the nilpotency index of \(B(K[S])\) and \(q\) the maximum of GK-dimensions of the cancellative subsemigroups of \(S\).
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 16P40
zbMATH DE Number: 554224
zbMATH Keywords: noetherian semigroup ring
zbMATH Keywords: semigroup algebra
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: semigroups with polynomial growth
zbMATH Keywords: prime radical
zbMATH Keywords: nilpotent-by-finite group of fractions
zbMATH Keywords: nilpotency index
zbMATH Keywords: cancellative subsemigroups
author: Jan Okniński
reviewed by: Josif S. Ponizovskii
MaRDI profile type: Publication
full work available at URL: jabr.1993.1255
OpenAlex ID: W2040211229
DOI: 10.1006/JABR.1993.1255
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description: scientific article; zbMATH DE number 554224