Modules with the summand intersection property. (English) Zbl 0667.16020

A unitary right module M over a ring R with identity is said to have the summand intersection property if the intersection of any two direct summands of M is a direct summand. The author characterizes all indecomposable R-modules A such that, for every (or for every finite) index set I, the module \(M=\oplus_ IA\) has the summand intersection property. She also characterizes the completely decomposable torsion-free abelian groups with the summand intersection property. Finally, she gives an example of a finite-rank torsion-free abelian group A and a subgroup B of finite index such that \(A\oplus A\oplus B\) does not have the summand intersection property while, for every index set I, the Z-module \(M=\oplus_ IA\) does.
Reviewer: T.W.Hungerford


16D80 Other classes of modules and ideals in associative algebras
20K15 Torsion-free groups, finite rank
20K25 Direct sums, direct products, etc. for abelian groups
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