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  "body": "\n## Summary\n\nA ring is stably finite when all its matrix rings are [[Dedekind-finite|GV1HKEAWVN]].\nBeing Dedekind-finite does not pass to matrix rings in general, so this is a stronger condition.\nStably finite nonzero rings have invariant basis number, so the rank of a finitely generated free module over them is well defined.\n\n## Explanations\n\n### L1 human\n\nA ring is Dedekind-finite when a one-sided inverse is always two-sided.\nStable finiteness asks the same thing for square matrices of every size over the ring.\nCommutative rings are stably finite: for square matrices over them, $AB = I$ gives $\\det(A)\\det(B) = 1$, and then $A$ is invertible.\nSo the rule \"a left inverse of a square matrix is a right inverse\" holds there in every size.\n",
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