Injectively homogeneous rings
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.
scientific article; zbMATH DE number 4057722
scientific article