Separative cancellation and multiple isomorphism in torsion-free Abelian groups
type: Item
instance of: scholarly article
title: Separative cancellation and multiple isomorphism in torsion-free Abelian groups
zbMATH Open document ID: 0945.20032
author: K. C. O'Meara
author: Charles Vinsonhaler
published in: Journal of Algebra
publication date: 2000-07-16T00:00:00Z
review text: Let all groups be Abelian and torsion-free of finite rank. Recall that a group \(A\) has separative cancellation if \(A\oplus A\cong A\oplus B\cong B\oplus B\) implies \(A\cong B\) and note that the problem of having separative cancellation generalizes the well-known cancellation problem and the \(n\)-th root problem. The group \(A\) is said to have the strong separative cancellation property if \(A\oplus A\cong A\oplus B\) implies \(A\cong B\). Finally recall that two groups \(A\) and \(B\) are multiple isomorphic if \(A^n\cong B^n\) for all \(n\geq 2\) holds. The authors show that for finite rank torsion-free groups separative cancellation, strong separative cancellation and multiple isomorphism are the same. Moreover, using theorems from number theory, they show that examples for the failure of strong separative cancellation are plentiful in the category of torsion-free Abelian groups of finite rank. Finally they show how multiple isomorphism fits into the hierachy of variations of isomorphism for torsion-free Abelian groups of finite rank and they construct in an explicit way families of examples of pairs \((A,B)\) of strongly indecomposable groups which are multiple isomorphic but not isomorphic. The construction relies on a number-theoretic conjecture on odd primes which was verified by computer for almost all primes below \(1000\). Nevertheless, the conjecture itself is still open.
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20K25
zbMATH DE Number: 1400106
zbMATH Keywords: multiple isomorphisms
zbMATH Keywords: strong separative cancellation
zbMATH Keywords: finite rank torsion-free Abelian groups
zbMATH Keywords: cancellation problem
zbMATH Keywords: strongly indecomposable groups
reviewed by: Lutz Strüngmann
MaRDI profile type: Publication
full work available at URL: jabr.1999.7990
OpenAlex ID: W2093700365
cites work: Separative cancellation for projective modules over exchange rings
cites work: Finite rank torsion free Abelian groups and rings
cites work: Endomorphism Rings and Direct Sums of Torsion Free Abelian Groups
cites work: Matrices, idealisers, and integer quaternions
cites work: Q3848243
cites work: Q5673086
cites work: Power-cancellation of groups and modules
cites work: The genus of a module. II: Roiter's theorem, power cancellation and extension of scalars
cites work: The l 1 -Algebra of a Commutative Semigroup
cites work: Nearly isomorphic torsion free abelian groups
cites work: Q3841752
cites work: Almost completely decomposable groups with two critical types
cites work: Q3961610
cites work: Q4061092
cites work: Projective modules over group rings and maximal orders
cites work: Simplification pour les ordres des corps de quaternions totalement définis.
cites work: Cancellation of modules and groups and stable range of endomorphism rings
cites work: Q3943047
cites work: Orders as Endomorphism Rings of Modules of the Same Rank
DOI: 10.1006/JABR.1999.7990
description: scientific article; zbMATH DE number 1400106