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  "body": "\n## Summary\n\nThe rank of an [[abelian group|082GHNPJ53]] is the largest number of its elements that are independent over the integers.\nIt equals the [[dimension|03GSZ8RTZJ]] of the rational vector space obtained by tensoring the group with $\\Q$.\nIt plays for abelian groups the role that dimension plays for vector spaces, but it ignores torsion: finite groups have rank $0$.\nFor a finitely generated abelian group, the rank is the number of copies of $\\Z$ in its decomposition.\n\n## Explanations\n\n### L1 human\n\nThe rank counts how many independent directions an abelian group has, the way $\\Z^n$ has $n$ coordinate directions.\nElements of finite order do not count as directions, so $\\Z \\oplus \\Z/2\\Z$ has rank $1$.\nThe group $\\Q$ also has rank $1$: any two nonzero rationals $a, b$ satisfy a relation $ma - nb = 0$ with integers $m, n$ not both zero.\nSo $\\Q$, though not finitely generated, has only one direction.\n",
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