Jónsson extensions of one-dimensional semilocal domains
type: Item
instance of: scholarly article
title: Jónsson extensions of one-dimensional semilocal domains
zbMATH Open document ID: 0654.13008
DOI: 10.1016/0021-8693(88)90249-9
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: A J-extension T of a commutative ring R is a commutative ring containing R, not finitely generated as an R algebra, but minimal with respect to this property, so that every R subalgebra of T unequal to T is finitely generated. The authors characterize the J-extensions T of a one- dimensional semilocal Noetherian domain R inside the field of fractions K of R: if \((.)^-\) denotes ``total-quotient-ring-of-completion'' then the minimal nonzero nilpotent ideals I of \(R^-\) correspond to such J- extensions T bijectively, with I the kernel of \(R^-\to T^-\) and T the intersection of K in R -/I with the completion R* of R modulo R*\(\cap I\). - This theorem also yields as a corollary the multiplicity mult(T) as mult(R)-mult(R*/P) (for P some minimal prime of R*). The authors further prove that if R is Gorenstein with local integral closure S, then there is at most one J-extension of R in K, and if also \(mult(R)=2\), then the rings between R and S are linarly ordered. Examples are given to show the hypotheses necessary.
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 4067057
zbMATH Keywords: Jónsson extensions of one-dimensional semilocal domains
zbMATH Keywords: J-extension
zbMATH Keywords: not finitely generated
zbMATH Keywords: multiplicity
zbMATH Keywords: Gorenstein
author: William J. Heinzer
author: David C. Lantz
reviewed by: Andy R. Magid
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90249-9
OpenAlex ID: W1978129325
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description: scientific article; zbMATH DE number 4067057