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Torsion-free abelian groups of rank 1
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Torsion-free abelian groups of rank 1 : ウィキペディア英語版
Torsion-free abelian groups of rank 1
Infinitely generated abelian groups have very complex structure and are far less well understood than finitely generated abelian groups. Even torsion-free abelian groups are vastly more varied in their characteristics than vector spaces. Torsion-free abelian groups of rank 1 are far more amenable than those of higher rank, and a satisfactory classification exists, even though there are an uncountable number of isomorphism classes.
==Definition==

A torsion-free abelian group of rank 1 is an abelian group such that every element except the identity has infinite order, and for any two non-identity elements ''a'' and ''b'' there is a non-trivial relation between them over the integers:
: n a + m b = 0 \;

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