Semi-local localizations of rings and subdirect decompositions of modules
type: Item
instance of: scholarly article
title: Semi-local localizations of rings and subdirect decompositions of modules
zbMATH Open document ID: 0624.16001
DOI: 10.1016/0022-4049(87)90090-9
author: Theodore G. Faticoni
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: The main purpose of the paper is the development of a non-commutative version of the classic Jonsson-Reid theorem which asserts that a torsionfree abelian group of finite rank is quasi-isomorphic to a finite direct sum of strongly indecomposable torsionfree abelian groups. The key concept that replaces the notion of quasi-isomorphism is the following: A pair of monomorphisms \(f: A\to A'\) and g: A'\(\to A\) in an additive category \({\mathcal M}\) is called a margimorphism pair in \({\mathcal M}\), and A and A' are called margimorphic, if fg and gf are regular maps, that is, nonzero divisors in \({\mathcal M}\). A direct sum \(B\oplus B'\) which is margimorphic to A is called a marginal decomposition of A. For an object A of the module category \({\mathcal M}\) whose endomorphism ring \(End_{{\mathcal M}}(A)\) has a classical right quotient ring \(Q_ A\), the finitely generated projective \(Q_ A\)-modules are described in terms of marginal decompositions of \(A^ n\). A major portion of the article is devoted to the development of the category theoretical machinery concerning margimorphisms, and many results are stated for additive categories.
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 18E05
Mathematics Subject Classification ID: 18A20
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16B50
zbMATH DE Number: 4012827
zbMATH Keywords: Jonsson-Reid theorem
zbMATH Keywords: quasi-isomorphism
zbMATH Keywords: monomorphisms
zbMATH Keywords: additive category
zbMATH Keywords: margimorphism
zbMATH Keywords: regular maps
zbMATH Keywords: marginal decomposition
zbMATH Keywords: module category
zbMATH Keywords: endomorphism ring
zbMATH Keywords: classical right quotient ring
zbMATH Keywords: finitely generated projective \(Q_ A\)-modules
zbMATH Keywords: margimorphisms
reviewed by: Günter Krause
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90090-9
OpenAlex ID: W1993302940
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description: scientific article; zbMATH DE number 4012827