LCM-stableness in ring extensions
type: Item
instance of: scholarly article
title: LCM-stableness in ring extensions
zbMATH Open document ID: 0531.13001
author: Hirohumi Uda
published in: Hiroshima Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: \textit{R. Gilmer} [in Ring Theory, Proc. Oklahoma Conf. 1973, Lect. Notes Pure Appl. Math. 7, 47-61 (1974; Zbl 0275.13019)] introduced the concept of LCM-stableness relating to GCD-properties of a commutative group ring. The main purpose of this paper is to point out that, in some cases, the necessary and sufficient conditions for a ring extension to be LCM-stable can be given in terms of polynomial grade, originally due to M. Hochster and developed by D. G. Northcott. Various different situations are considered and we shall attempt to give only a flavour of the results obtained. To be more explicit we fix the following notation. Let \(A\subset B\) be integral domains, K the quotient field of A and \(\Omega\) the algebraic closure of K. If I is an ideal of A let Gr(I) denote the polynomial grade of I. Let X denote an indeterminate and for any ideal J of the polynomial ring A[X], let c(J) denote the ideal of A generated by all coefficients of all polynomials in J. Finally if \(\alpha\in \Omega\) let \(A_{\alpha}\) denote the kernel of the canonical homomorphism of A[X] onto A[\(\alpha]\). Then \(A\subset B\) is called LCM-stable provided \((aA\cap bA)B=aB\cap bB\) for all a,b in A. In particular, if \(A\subset B\) is flat then \(A\subset B\) is LCM-stable, and the converse is true if B is a subring of K. Moreover, it is easily proved that \(A\subset B\) is LCM- stable provided \(A\subset A[x,y]\) is LCM-stable for all x,y in B. Now let 0\(\neq \alpha \in \Omega\) with \(\alpha^ m=u\in K\) for some positive integer m. Put \(u=a/b\) where 0\(\neq a,b\in A\) and assume that \(bX^ m-a\) is irreducible over K and \(A_ u\) has a linear base. Then \(A\subset A[\alpha]\quad is\quad LCM-stable\Leftrightarrow A\subset A[\alpha]\quad is\quad INC\Leftrightarrow A\subset A[\alpha]\quad is\quad flat\Leftrightarrow(a,b)\quad is\quad invertible.\) The author introduces two further ideas namely \(G_ 2\)-stableness and \(R_ 2\)-stableness and shows how these are related to LCM-stable…
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F15
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 3842017
zbMATH Keywords: LCM-stable extension of integral domains
zbMATH Keywords: \(G_ 2\)-stable extension
zbMATH Keywords: \(R_ 2\)-stable extension
zbMATH Keywords: polynomial ring extension
zbMATH Keywords: polynomial grade
zbMATH Keywords: GCD-domain
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3842017