About composite rings
type: Item
instance of: scholarly article
zbMATH Open document ID: 0901.13017
author: Gabriel Picavet
publication date: 1998-12-02T00:00:00Z
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13F45
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13C14
zbMATH DE Number: 983089
zbMATH Keywords: composite rings
zbMATH Keywords: locally noetherian, non-noetherian ring
zbMATH Keywords: seminormalization
zbMATH Keywords: Cohen-Macaulay
MaRDI profile type: Publication
title: About composite rings
review text: Let \(A\) be a subring of a ring \(B\). The composite ring of the pair \((A,B)\) is \(R: =A+ XB[X]\). The paper is devoted to a study of composite rings, which are very good tools for (counter)examples. Browsing through the given results, one can choose some of them as follows.NEWLINENEWLINENEWLINEA factor ring of a composite ring \(R\) is the idealization of \(B\) in \(A\) and there is a natural surjective morphism from the symmetric algebra \(S_A(B)\) to \(R\). Flatness criteria for composite ring morphisms are given. If \(I\) is the ideal \(XB [X]\) of \(R\), properties of \(I\) and \(R\) are closely related. For example, \(I\) is finitely generated if and only if \(A\to B\) is finite. This enables the author to prove that \(R\) is noetherian (resp. locally noetherian) if and only if \(A\to B\) is finite (resp. locally finite) and \(A\) and \(B\) are noetherian (resp. locally noetherian). An example of a locally noetherian, non-noetherian ring is thus obtained. Arithmetical properties of composite rings are also studied using normal pairs. The seminormalization of a composite ring is computed. Finally, if \(A\to B\) is flat and finite, it is shown that \(R\) is Cohen-Macaulay if and only if \(A\) and \(B\) are Cohen-Macaulay.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].
reviewed by: Cristodor-Paul Ionescu
description: scientific article; zbMATH DE number 983089