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Modules with the quasi-summand intersection property

Published online by Cambridge University Press:  17 April 2009

Ulrich Albrecht  and
Jutta Hausen
Ulrich Albrecht
Affiliation:
Department of Mathematics, Auburn University, Auburn AL 36849-5307, United States of America
Jutta Hausen
Affiliation:
Department of Mathematics, University of Houston, Houston TX 77204-3476, United States of America
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Abstract

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Given a torsion-free abelian group G, a subgroup A of G is said to be a quasi-summand of G if nG ≤ A ⊕ B ≤ G for some subgroup B of G and some positive integer n. If the intersection of any two quasi-summands of G is a quasi-summand, then G is said to have the quasi-summand intersection property. This is a generalisation of the summand intersection property of L. Fuchs. In this note, we give a complete characterisation of the torsion-free abelian groups (in fact, torsion-free modules over torsion-free rings) with the quasi-summand intersection property. It is shown that such a characterisation cannot be given via endomorphism rings alone but must involve the way in which the endomorphism ring acts on the underlying group. For torsion-free groups G of finite rank without proper fully invariant quasi-summands however, the structure of its quasi-endomorphism ring QE(G) suffices: G has the quasi-summand intersection property if and only if the ring QE(G) is simple or else G is strongly indecomposable.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 2 , October 1991 , pp. 189 - 201
Copyright
Copyright © Australian Mathematical Society 1991

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