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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
Full Text: DOI
[1] Albrecht U., Lecture Notes in Math. 1006 pp 209– (1983)
[2] Albrecht U., Trans. Amer. Math. Soc. 293 pp 565– (1986)
[3] Arnold D.M., Lecture Notes in Math. 931 (1982)
[4] DOI: 10.1090/S0002-9947-1975-0417314-1 · doi:10.1090/S0002-9947-1975-0417314-1
[5] Arnold D.M., Pacific J. Math. 56 pp 7– (1975)
[6] DOI: 10.1080/00927878308822908 · Zbl 0515.20034 · doi:10.1080/00927878308822908
[7] Bowman H., Houston J. Math. 11 pp 447– (1985)
[8] Fuchs L., Infinite Abelian Groups (1970) · Zbl 0209.05503
[9] Fuchs L., Infinite Abelian Groups (1973) · Zbl 0257.20035
[10] DOI: 10.1090/S0002-9939-1985-0770526-1 · doi:10.1090/S0002-9939-1985-0770526-1
[11] Huber M., Lecture Notes in Math. 874 (1981)
[12] Hungerford T.W., Algebra (1974) · Zbl 0293.12001
[13] DOI: 10.2307/1970252 · Zbl 0083.25802 · doi:10.2307/1970252
[14] Kaplansky I., Infinite Abelian Groups (1969) · Zbl 0194.04402
[15] Rotman J.J., An Introduction to Homological Algebras (1979) · Zbl 0441.18018
[16] DOI: 10.1080/00927878608823297 · Zbl 0592.13008 · doi:10.1080/00927878608823297
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