Finitely generated intermediate rings
type: Item
instance of: scholarly article
title: Finitely generated intermediate rings
zbMATH Open document ID: 0564.13009
DOI: 10.1016/0022-4049(85)90101-X
author: Robert Gilmer
author: William J. Heinzer
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let R be a commutative unitary ring and let S be an affine extension of R. If each intermediate ring T, \(R\subseteq T\subseteq S\), is affine over R, then S is said to be strongly affine over R. This condition implies that tr.deg.S/R\(\leq 1\), and if equality holds, then R is Noetherian and von Neumann regular. This paper investigates various questions concerning strongly affine extensions. Primary emphasis is on the case where R is Noetherian, in which case (R,S) is a Noetherian pair in the sense of \textit{A. R. Wadsworth} [Trans. Am. Math. Soc. 195, 201-211 (1974; Zbl 0294.13010)]. While the converse fails in general, it is shown that if R is a Noetherian universally Japanese ring, if S/R is affine, and if (R,S) is a Noetherian pair, then S is strongly affine over R. One section of the paper is devoted to a consideration of the strongly affine condition in the case where R is an integral domain and S is an overring of R, particularly for the case where R is a Prüfer domain. For example, if R is a Prüfer domain satisfying a.c.c. on prime ideals, then each affine overring of R is strongly affine over R if and only if each nonzero ideal of R is contained in only finitely many prime ideals of R. Conversely, an integrally closed domain R is a Prüfer domain if each affine overring is strongly affine over R. However, the following question remains open: If each affine overring of the domain D is strongly affine over D, is the integral closure of D a Prüfer domain?
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 3899041
zbMATH Keywords: intermediate ring
zbMATH Keywords: strongly affine extensions
zbMATH Keywords: Japanese ring
zbMATH Keywords: Noetherian pair
zbMATH Keywords: Prüfer domain
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3899041