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  "body": "\n## Definition\n\nLet $A$ be an [[abelian group|082GHNPJ53]], written additively.\nA basis of $A$ is a subset $B \\subseteq A$ such that each $a \\in A$ has exactly one expression $a = k_1 b_1 + \\cdots + k_n b_n$ with distinct $b_i \\in B$ and nonzero integers $k_i$.\nThe group $A$ is a free abelian group when it has a basis.\n\n## Remarks\n\nThe additive group $\\mathbb{Z}$ is free abelian with basis $\\{1\\}$, and $\\mathbb{Z}^2$ is free abelian with basis $\\{(1, 0), (0, 1)\\}$.\nAll bases of $A$ have the same cardinality, the rank, and a free abelian group of finite rank $n$ is [[isomorphic|08MG7H8JW5]] to $\\mathbb{Z}^n$.\nThe free abelian group on a set $S$ is the [[abelianisation|0AQZV0905N]] of the [[free group|0KTW26C80J]] $F_S$.\nEach subgroup of a free abelian group is free abelian.\n\n## Explanations\n\n### L1 human\n\nA free abelian group is like a vector space over the integers with a basis.\nEach element is an integer combination of basis elements in exactly one way.\nThe standard examples are $\\mathbb{Z}^n$ for $n \\geq 0$.\n",
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