A note on torsion-free abelian groups of finite rank

Authors:
W. Wickless and C. Vinsonhaler

Journal:
Proc. Amer. Math. Soc. **39** (1973), 63-68

MSC:
Primary 20K15

DOI:
https://doi.org/10.1090/S0002-9939-1973-0313419-8

MathSciNet review:
0313419

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Abstract: Let be a torsion-free abelian group of rank and a maximal set of rationally independent elements in . It is well known that any can be uniquely written , for some , the rational numbers. This enables us to define, for any such , a collection of sub-groups of and ``natural'' isomorphisms, denoted by . It is known that if is of rank two, then may be recovered from is a natural way. The following result is obtained for groups of rank greater than two:

Theorem. *Let be torsion free abelian groups of finite rank with for suitable . Let be the free subgroups of generated by . Then *.

*An additional condition is given for pairs ( such that implies .*

**[1]**R. A. Beaumont and R. S. Pierce,*Torsion-free rings*, Illinois J. Math.**5**(1961), 61–98. MR**0148706****[2]**R. A. Beaumont and R. J. Wisner,*Rings with additive group which is a torsion-free group of rank two.*, Acta Sci. Math. Szeged**20**(1959), 105–116. MR**0106921**

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DOI:
https://doi.org/10.1090/S0002-9939-1973-0313419-8

Article copyright:
© Copyright 1973
American Mathematical Society