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  "body": "\n## Summary\n\nA ring has invariant basis number (IBN) if any two bases of a finitely generated [[free module|0DZ9DKAKZR]] over it have the same size.\nFor such a ring the rank of a free module is well defined, just as the dimension of a vector space is.\nAll nonzero commutative rings and all division rings have this property, but some noncommutative rings do not.\n\n## Explanations\n\n### L1 human\n\nOver a field, every basis of a vector space has the same number of elements; this number is the dimension.\nFor modules over a general ring this can fail: a free module can have a basis of $1$ element and also a basis of $2$ elements.\nA ring has invariant basis number when this strange behaviour does not happen.\nFor example, $\\Z$ has invariant basis number, so $\\Z^m$ and $\\Z^n$ are isomorphic only when $m = n$.\n",
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