GK-dimensions of corners and ideals
type: Item
instance of: scholarly article
title: GK-dimensions of corners and ideals
zbMATH Open document ID: 0697.16025
DOI: 10.1007/BF02937301
author: Lance W. Small
author: S. A. Amitsur
published in: Israel Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Let R be a prime algebra over a field k, and let e be an idempotent element of R. If R is either PI or noetherian, then it is known (see, for example, \textit{S. Montgomery} [Contemp. Math. 88, 103-124 (1989; Zbl 0675.16018)]) that \(GK(eRe)=GK(R)\), where GK(-) denotes the Gelfand- Kirillov dimension, and the problem remains open for the prime Goldie case. The authors give an example to show that in a general prime ring GK(eRe) may be any integer \(<GK(R)\). The key result for the analysis is that for a corner (also called subideal) of R, that is, a subalgebra T with TRT\(\subseteq T\), one always has \(GK(T)=GK(RT^ 2R)\). The above- stated problem is closely connected with the problem whether \(GK(R)=GK(A)\), as an algebra, for any nonzero ideal A of a prime k- algebra R. In fact, for prime Goldie rings the two questions are equivalent, meaning that a prime Goldie ring R exists with a nonzero ideal A such that \(GK(A)<GK(R)\), if and only if a prime Goldie ring S exists with an idempotent e such that \(GK(eSe)<GK(S)\). On the positive side, it is shown that if R is prime Goldie with GK(R)\(\leq 2\), then indeed \(GK(A)=GK(R)\) for any ideal \(A\neq 0\).
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 4142254
zbMATH Keywords: prime algebra
zbMATH Keywords: idempotent element
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: prime ring
zbMATH Keywords: corner
zbMATH Keywords: subideal
zbMATH Keywords: prime Goldie rings
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4142254