Note on direct decompositions of torsion-free abelian groups

1971 ◽  
Vol 46 (1) ◽  
pp. 409-413 ◽  
Author(s):  
L. Fuchs
Keyword(s):  
Abelian Groups ◽  
Free Abelian Groups ◽  
Direct Decompositions ◽  
Torsion Free

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

1961 ◽  
Vol 57 (2) ◽  
pp. 230-233 ◽  
Author(s):  
A. L. S. Corner
Keyword(s):  
Abelian Groups ◽  
Free Abelian Groups ◽  
Positive Integers ◽  
Direct Decompositions ◽  
Torsion Free

Fuchs((1), Problem 22) has asked the following question: Given positive integers ri (1 ≤ i ≤ 4) such thatr1 + r2 = r3 + r4, r1 ≠ r3, r1 ≠ r4, do there exist indecomposable torsion-free Abelian groups Gi (1 ≤ i ≤ 4) such that, where Gi is of rank ri (1 ≤ i ≤ 4)?* We shall show in this note that there do always exist groups Gi with the desired properties; in fact, we shall prove the following stronger result

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

1969 ◽  
Vol 66 (2) ◽  
pp. 239-240 ◽  
Author(s):  
A. L. S. Corner
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Direct Sums ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Direct Decompositions ◽  
Torsion Free

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 ℵ0 and as a direct sum G = ⊕n ε zCn of ℵ0 indecomposable groups Cn (nεZ) of rank 2, where Z is the set of all integers.

On the square subgroups of decomposable torsion-free abelian groups of rank three

2016 ◽  
Vol 7 (4) ◽  
Author(s):  
Fysal Hasani ◽  
Fatemeh Karimi ◽  
Alireza Najafizadeh ◽  
Yousef Sadeghi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractThe square subgroup of an abelian group

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