Homological properties of generic matrix rings
type: Item
instance of: scholarly article
title: Homological properties of generic matrix rings
zbMATH Open document ID: 0581.16013
DOI: 10.1007/BF02772956
published in: Israel Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Although generic matrix rings play a very important role in the theory of rings with polynomial identity, little is known about their homological properties. It is suspected that their global dimension is always infinite, and the authors establish this for the ring \(R=k_{(2)}\{a,b\}\) of two generic \(2\times 2\) matrices over an infinite field k. The trace ring T of R is shown to be an iterated Ore extension with \(gl.\dim (T)=5\), to be a maximal order, and to be stably free. Furthermore, if \(A=R\) or T, then there exists a non-free projective right ideal P satisfying \(P\oplus A\simeq A\oplus A\). It is announced that R is also stably free, a proof will be furnished by Severino Collier Coutinho in a forthcoming paper.
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16Exx
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 3929206
zbMATH Keywords: generic matrix rings
zbMATH Keywords: rings with polynomial identity
zbMATH Keywords: global dimension
zbMATH Keywords: generic 2\(\times 2\) matrices
zbMATH Keywords: trace ring
zbMATH Keywords: iterated Ore extension
zbMATH Keywords: maximal order
zbMATH Keywords: stably free
author: Lance W. Small
author: J. T. Stafford
reviewed by: Günter Krause
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3929206