On the \(GE_ 2\) of graded rings
type: Item
instance of: scholarly article
title: On the \(GE_ 2\) of graded rings
zbMATH Open document ID: 0541.20033
DOI: 10.1016/0021-8693(84)90207-2
author: Huah Chu
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: A ring R is said to be a G\(E_ 2\)-ring if \(GL_ 2(R)\), the group of 2\(\times 2\) invertible matrices over R, is generated by elementary and diagonal matrices. The author obtains some necessary conditions for a graded domain to be a \(GE_ 2\)-ring. This gives rise to a number of interesting consequences. Amongst other things it is proved, for example, that the polynomial ring D[X] over a commutative Noetherian ring D is a \(GE_ 2\)-ring iff every prime ideal in D is maximal (i.e. D is Artinian). Another result of interest is that the Laurent polynomial ring \(D[X,X^{-1}]\) over a commutative valuation domain D is a \(GE_ 2\)- ring. (Reviewer's remark: As far as I know it is not known if the Laurent polynomial ring over the rational integers is a \(GE_ 2\)-ring.)
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 3859404
zbMATH Keywords: G\(E_ 2\)-ring
zbMATH Keywords: diagonal matrices
zbMATH Keywords: graded domain
zbMATH Keywords: Laurent polynomial ring
zbMATH Keywords: commutative valuation domain
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90207-2
OpenAlex ID: W2033684125
cites work: Q3903156
cites work: Q5515450
cites work: On the structure of the \(GL_ 2\) of a ring
cites work: The construction \(D+XD_s[X]\)
description: scientific article; zbMATH DE number 3859404