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A note on rank and direct decompositions of torsion-free Abelian groups. II

1969; Cambridge University Press; Volume: 66; Issue: 2 Linguagem: Inglês

10.1017/s0305004100044911

1469-8064

A. L. S. Corner,

graph theory and CDMA systems

According to well-known theorems of Kaplansky and Baer–Kulikov–Kapla nsky–Fuchs (4, 2), the class of direct sums of countable Abelian groups and the class of direct sums of torsion-free Abelian groups of rank 1 are both closed under the formation of direct summands. In this note I give an example to show that the class of direct sums of torsion-free Abelian groups of finite rank does not share this closure property: more precisely, there exists a torsion-free Abelian group G which can be written both as a direct sum G = A⊕B of 2 indecomposable groups A, B of rank ℵ 0 and as a direct sum G = ⊕ n ε z C n of ℵ 0 indecomposable groups C n (nεZ) of rank 2 , where Z is the set of all integers.