A note on torsion-free abelian groups of finite rank
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- by W. Wickless and C. Vinsonhaler PDF
- Proc. Amer. Math. Soc. 39 (1973), 63-68 Request permission
Abstract:
Let $G$ be a torsion-free abelian group of rank $n$ and $X = \{ {x_1}, \cdots ,{x_n}\}$ a maximal set of rationally independent elements in $G$. It is well known that any $g \in G$ can be uniquely written $g = {\alpha _1}{x_1} + \cdots + {\alpha _n}{x_n}$, for some ${\alpha _1}, \cdots ,{\alpha _n} \in Q$, the rational numbers. This enables us to define, for any such $(G,X)$, a collection of sub-groups of $Q$ and “natural” isomorphisms, denoted by $S(G,X)$. It is known that if $G$ is of rank two, then $G$ may be recovered from $S(G,X)$ is a natural way. The following result is obtained for groups of rank greater than two: Theorem. Let $G,G’$ be torsion free abelian groups of finite rank with $S(G,X) = S(G’,X’)$ for suitable $X,X’$. Let $F,F’$ be the free subgroups of $G,G’$ generated by $X,X’$. Then $G/F \cong G’/F’$. An additional condition is given for pairs ($(G,X),(G’,X’)$ such that $S(G,X) = S(G’,X’)$ implies $G \cong G’$.References
- R. A. Beaumont and R. S. Pierce, Torsion-free rings, Illinois J. Math. 5 (1961), 61–98. MR 148706
- 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 106921
Additional Information
- © Copyright 1973 American Mathematical Society
- 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