The Exchange Property of Modules with the Finite Exchange Property

Dhompongsa, Sompong; Tansee, Hansuk

doi:10.1081/agb-120017347

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…However, this has been proved for various classes of modules, including, for instance, modules that are direct sums of indecomposables ( [89]), and quasicontinuous modules ( [70], [67]). A positive answer was also recently given, in [27], for modules with abelian endomorphism rings. There seems to be an error in this paper, but the result has been reaffirmed by a different, and much shorter, proof by P. P. Nielsen [69].…”

Section: Epiloguementioning

confidence: 90%

A Crash Course on Stable Range, Cancellation, Substitution and Exchange

Lam

2004

J. Algebra Appl.

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The themes of cancellation, internal cancellation, substitution and exchange have led to a lot of interesting research in the theory of modules over commutative and noncommutative rings. This article provides a quick and relatively self-contained introduction to the voluminous work in this area, using the notion of the stable range of rings as a unifying tool. With only a small number of exceptions, all theorems stated here are proved in full, modulo basic facts in the theory of modules and rings available in standard textbooks on ring theory.

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Section: Epiloguementioning

confidence: 90%

A Crash Course on Stable Range, Cancellation, Substitution and Exchange

Lam

2004

J. Algebra Appl.

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“…It is proved in [2] that any weak duo module which satisfies the finite exchange property satisfies the (unrestricted) exchange property.…”

Section: ])mentioning

confidence: 99%

Duo Modules

Özcan¹,

Harmanci²,

SMITH³

2006

Glasgow Math. J.

104

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Abstract. Let R be a ring. An R-module M is called a (weak) duo module provided every (direct summand) submodule of M is fully invariant. It is proved that if R is a commutative domain with field of fractions K then a torsion-free uniform R-module is a duo module if and only if every element k in K such that kM is contained in M belongs to R. Moreover every non-zero finitely generated torsion-free duo R-module is uniform. In addition, if R is a Dedekind domain then a torsion R-module is a duo module if and only if it is a weak duo module and this occurs precisely when the P-primary component of M is uniform for every maximal ideal P of R.2000 Mathematics Subject Classification. 16D99, 13C12 (13B20). The right R-module M is called a duo module provided every submodule of M is fully invariant. For example, if U is a simple right R-module, then clearly U is a duo module but U ⊕ U is not duo. The ring R is called a right duo ring if the right R-module R is a duo module. Note that a ring R is a right duo ring if and only if every right ideal of R is a two-sided ideal; equivalently Ra is contained in aR for every element a in R. Clearly commutative rings and division rings are right (and left) duo rings but any 2 × 2 matrix ring over such a ring is not a right (or left) duo ring.We begin with a simple observation.

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Abelian Exchange Modules

Nielsen

2005

Communications in Algebra

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The Exchange Property of Modules with the Finite Exchange Property

Cited by 3 publications

References 6 publications

A Crash Course on Stable Range, Cancellation, Substitution and Exchange

A Crash Course on Stable Range, Cancellation, Substitution and Exchange

Duo Modules

Abelian Exchange Modules

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