Ideals of finite codimension in free algebras and the \(fc\)-localization
type: Item
instance of: scholarly article
title: Ideals of finite codimension in free algebras and the \(fc\)-localization
zbMATH Open document ID: 0795.16020
author: Amnon Rosenmann
author: Shmuel Rosset
published in: Pacific Journal of Mathematics
publication date: 1994-07-19T00:00:00Z
review text: The topology defined by all finite codimensional right ideals of an algebra over a field is an analogue of the profinite topology of group theory. In the case of the free algebra and the group ring of a free group it has many interesting properties. For example, we show that its open ideals are precisely the finitely generated essential ones. Also, finitely generated right ideals are closed. This topology is what is called a ``Gabriel'' topology, and following Gabriel's general procedure it gives rise to a localization of the algebra which we call the \(fc\)- localization. In the free case this localization does not have the unique rank property and we show that its ``Leavitt'' numbers are 1 and \(n-1\). This means that two free modules over it are isomorphic if and only if their ranks differ by a multiple of \(n-1\). The proofs are, for the most part, applications of varying complexity of Schreier's method.
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 537389
zbMATH Keywords: Gabriel topology
zbMATH Keywords: Leavitt numbers
zbMATH Keywords: finite codimensional right ideals
zbMATH Keywords: profinite topology
zbMATH Keywords: free algebra
zbMATH Keywords: group ring
zbMATH Keywords: free group
zbMATH Keywords: open ideals
zbMATH Keywords: finitely generated right ideals
zbMATH Keywords: localization
zbMATH Keywords: \(fc\)-localization
zbMATH Keywords: free modules
zbMATH Keywords: ranks
zbMATH Keywords: Schreier's method
MaRDI profile type: Publication
full work available at URL: pjm.1994.162.351
OpenAlex ID: W1992726658
DOI: 10.2140/PJM.1994.162.351
description: scientific article; zbMATH DE number 537389