Power cancellation of modules
type: Item
instance of: scholarly article
title: Power cancellation of modules
zbMATH Open document ID: 0594.13007
author: Robert Guralnick
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let Mod-R (resp. \(Mod_ f-R)\) denote the category of unital right (resp. finitely generated) R-modules. \(X\in Mod_ f-R\) satisfies ''power cancellation'' if for all M,N\(\in Mod-R\), \(M\oplus X\cong N\oplus X\Rightarrow M^{(t)}\cong N^{(t)}\) for some \(0<t\in {\mathbb{Z}}\). Here \(M^{(t)}\) is the direct sum of t copies of M. If D is a Dedekind domain, any finitely generated module satisfies cancellation (i.e., \(t=1)\). If R is a \({\mathbb{Z}}\)-order in a separable \({\mathbb{Q}}\)-algebra, then power cancellation holds for R-lattices. It is shown in this article that for R being an integrally closed domain, \(M\oplus X\cong N\oplus X\Rightarrow M^{(t)}\cong N^{(t)}\) for some \(0<t\in {\mathbb{Z}}\) for all finitely generated S-modules M, N, X, whenever S is a module finite algebra if and only if one is in the stable range of the integral closure of R in the algebraic closure of its quotient field. In particular, this holds whenever R is a Dedekind domain with all residue fields torsion. This extends work of Goodearl, who showed this holds for module finite (and, more generally, finite rank) algebras over the integers.
Mathematics Subject Classification ID: 13C05
zbMATH DE Number: 3955063
zbMATH Keywords: power cancellation
zbMATH Keywords: integral closure
zbMATH Keywords: Dedekind domain
MaRDI profile type: Publication
full work available at URL: pjm.1986.124.131
OpenAlex ID: W2120388241
DOI: 10.2140/PJM.1986.124.131
description: scientific article; zbMATH DE number 3955063