Connected graded Gorenstein algebras with enough normal elements
type: Item
instance of: scholarly article
title: Connected graded Gorenstein algebras with enough normal elements
zbMATH Open document ID: 0937.16049
author: J. J. Zhang
published in: Journal of Algebra
publication date: 2000-06-04T00:00:00Z
review text: Let \(A=k\oplus A_1\oplus A_2\oplus\cdots\) be a connected graded algebra. The algebra \(A\) is said to have enough normal elements if for every graded prime ideal \(P\) of \(A\) the factor ring \(A/P\) contains a nonzero homogeneous normal element of degree \(\geq 1\). The author proves the following theorem which generalises earlier work of \textit{J. T. Stafford} and the author [J. Algebra 168, No. 3, 988-1026 (1994; Zbl 0812.16046)]. Theorem: Let \(A\) be a connected graded Noetherian algebra of injective dimension \(d\) with enough normal elements. Then (i) \(A\) is Auslander-Gorenstein and Cohen-Macaulay, (ii) \(A\) has a quasi-Frobenius quotient ring, (iii) every minimal prime ideal \(P\) is graded and \(\text{GKdim}(A/P)=d\), (iv) if, moreover, \(A\) has finite global dimension, then \(A\) is a domain and a maximal order in its quotient division ring. The earlier work of Stafford and Zhang applies to PI rings, which certainly have enough normal elements. Ample justification for this generalization is provided by the fact that many quantum algebras, while not being PI rings nevertheless have enough normal elements [see \textit{K. R. Goodearl} and the reviewer, J. Pure Appl. Algebra 111, No.~1-3, 123-142 (1996; Zbl 0864.16018)].
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16E65
zbMATH DE Number: 1013991
zbMATH Keywords: connected graded algebras
zbMATH Keywords: graded prime ideals
zbMATH Keywords: homogeneous normal elements
zbMATH Keywords: graded Noetherian algebras
zbMATH Keywords: injective dimensions
zbMATH Keywords: enough normal elements
zbMATH Keywords: quasi-Frobenius quotient rings
zbMATH Keywords: global dimensions
zbMATH Keywords: maximal orders
zbMATH Keywords: quantum algebras
reviewed by: T. H. Lenagan
MaRDI profile type: Publication
full work available at URL: jabr.1996.6885
OpenAlex ID: W2087070452
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DOI: 10.1006/JABR.1996.6885
description: scientific article; zbMATH DE number 1013991