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  "body": "\n## Summary\n\nA ring is Dedekind-finite when every one-sided inverse is a two-sided inverse: $ab = 1$ forces $ba = 1$.\nSuch rings are also called directly finite.\nCommutative rings, finite rings and matrix rings over fields are Dedekind-finite, while rings of operators on infinite-dimensional spaces need not be.\n\n## Explanations\n\n### L1 human\n\nFor square matrices over a field, if $AB = I$ then also $BA = I$.\nA ring with this property is Dedekind-finite.\nThe name recalls Dedekind's definition of a finite set: a set is finite when it is not in bijection with a proper subset.\nConsider the ring of linear maps on the space of infinite sequences.\nThere, \"shift right\" followed by \"shift left\" is the identity.\nBut \"shift left\" followed by \"shift right\" loses the first term, so that ring is not Dedekind-finite.\n",
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