Marginal isomorphisms.
type: Item
instance of: scholarly article
title: Marginal isomorphisms.
zbMATH Open document ID: 1147.16005
author: Theodore G. Faticoni
published in: Journal of Algebra
publication date: 2008-07-29T00:00:00Z
review text: The classical right ring of quotients of a ring \(R\) is a superring where the regular elements (i.e. which are not zero divisors) of \(R\) are invertible, and all the elements of the superring are of the form \(xc^{-1}\) where \(x\) and \(c\) are elements of \(R\) and \(c\) is regular in \(R\). (Such a superring does not always exist.) Two modules \(G\) and \(H\) are margimorphic if there are maps \(a\colon G\to H\), \(b\colon H\to G\) such that the compositions \(ab\) and \(ba\) are regular (not zero divisors) in the respective endomorphism rings. A marginal summand is a direct summand of a margimorphic copy. The article develops a whole theory of marginal summands of finite direct sums of copies of a module \(G\) whose endomorphism ring has a classical right ring of quotients, and this quotient ring is semi-primary. The culmination of the theory is that margimorphism classes of marginal summands are in one-to-one correspondence with finitely generated projective modules over the quotient ring (Theorem 8.1). This correspondence preserves direct sums, leading to an Azumaya-Krull-Schmidt type decomposition theorem: marginal summands of \(G\) decompose uniquely to totally indecomposable modules (indecomposable modules up to margimorphism) up to margimorphism (Theorem 11.2). When the factor of the quotient ring with the Jacobson radical is a product of division rings, then the module uniquely decomposes into a finite direct sum of indecomposable modules (Corollary 13.7). The final section of the article contains classifications of modules whose endomorphism ring has a semi-primary classical right ring of quotients. A sample result is: the endomorphism ring has a semi-simple Artinian classical right ring of quotients if and only if the module is margimorphic to a finite direct sum of copies of some modules \(K_i\), such that \(\Hom(K_i,K_j)=0\) for \(i\neq j\), and the endomorphism ring of \(K_i\) has a classical right ring of quotients which is a division ring (Theore…
reviewed by: Gábor Braun
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16U20
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 5304626
zbMATH Keywords: margimorphisms
zbMATH Keywords: marginal direct summands
zbMATH Keywords: Azumaya-Krull-Schmidt theorem
zbMATH Keywords: classical right quotient rings
zbMATH Keywords: semi-primary rings
zbMATH Keywords: endomorphism rings
zbMATH Keywords: decomposition theorems
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.02.014
OpenAlex ID: W4206489523
cites work: Modules with decompositions that complement direct summands
cites work: Finite rank torsion free Abelian groups and rings
cites work: Q4101984
cites work: Orders in Simple Artinian Rings
cites work: Q3428696
cites work: Direct sums and refinement
cites work: Semi-local localizations of rings and subdirect decompositions of modules
cites work: Q3717234
cites work: On Quasi-Decompositions of Torsion Free Abelian Groups
DOI: 10.1016/J.JALGEBRA.2008.02.014
description: scientific article; zbMATH DE number 5304626