Injectively homogeneous rings
type: Item
instance of: scholarly article
title: Injectively homogeneous rings
zbMATH Open document ID: 0648.16017
DOI: 10.1016/0022-4049(88)90078-3
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let R be a Noetherian ring integral over a central subring \(\Sigma\). For any ideal I of R, define the upper grade \(u.gr(I)=\sup \{n:\) Ext n(R/I,R)\(\neq 0\}\). The ring R is said to be right injectively homogeneous over \(\Sigma\) provided R has finite right injective dimension and \(u.gr(M_ R)=u.gr(M_ R')\) for all maximal ideals M, M' of R with \(M\cap \Sigma =M'\cap \Sigma\). The authors propose this as the natural extension to rings integral over their centres of the class of commutative Gorenstein rings. In support of this proposal, they show that most of the basic properties of Gorenstein rings in \textit{H. Bass}' papers [Trans. Am. Math. Soc. 102, 18-29 (1962; Zbl 0126.065); Math. Z. 82, 8-28 (1963; Zbl 0112.266)] can be established for these rings. The authors work supersedes cases considered by \textit{M. Ramras} [Trans. Am. Math. Soc. 142, 457-479 (1969; Zbl 0186.071)] and \textit{W. Vasconcelos} [Symp. Math. 11, 11-22 (1973; Zbl 0273.13013)]. In addition the authors link their work to their earlier consideration of homologically homogeneous rings [Trans. Am. Math. Soc. 281, 197-208 (1984; Zbl 0531.16019)], for a ring R is homologically homogeneous if and only if it is injectively homogeneous and has finite global dimension. They show that both injective (and thus homological) homogeneity holds on the right if and only if it holds on the left. Finally the authors provide examples of such rings which are polynomial rings or group rings, in the latter case characterizing them.
Mathematics Subject Classification ID: 16Exx
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16P40
zbMATH DE Number: 4057722
zbMATH Keywords: Noetherian ring
zbMATH Keywords: upper grade
zbMATH Keywords: right injectively homogeneous
zbMATH Keywords: finite right injective dimension
zbMATH Keywords: maximal ideals
zbMATH Keywords: rings integral over their centres
zbMATH Keywords: Gorenstein rings
zbMATH Keywords: homologically homogeneous rings
zbMATH Keywords: global dimension
zbMATH Keywords: polynomial rings
zbMATH Keywords: group rings
author: Kenneth A. Brown
author: C. R. Hajarnavis
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90078-3
OpenAlex ID: W2034057174
cites work: Injective Dimension in Noetherian Rings
cites work: On the ubiquity of Gorenstein rings
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description: scientific article; zbMATH DE number 4057722