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  "body": "\n## Definition\n\nLet $R$ be a ring with $1$.\nFor $n \\ge 1$, let $M_n(R)$ be the [[matrix ring|0FHVPRAE60]] of $n \\times n$ matrices over $R$.\nThe ring $R$ is *stably finite* if $M_n(R)$ is [[Dedekind-finite|GV1HKEAWVN]] for every $n \\ge 1$.\nThat is, for all $n \\ge 1$ and $A, B \\in M_n(R)$, the equation $AB = I_n$ implies $BA = I_n$.\n\n## Remarks\n\nExample: every commutative ring $R$ is stably finite.\nIf $AB = I_n$ in $M_n(R)$, then $\\det(A) \\det(B) = 1$, so $A$ has the inverse $\\det(A)^{-1} \\operatorname{adj}(A)$, and $BA = A^{-1}(AB)A = I_n$.\n\nNon-example: the ring $\\End(V)$ of linear maps on an infinite-dimensional vector space $V$ is not stably finite, since already $M_1(\\End(V)) = \\End(V)$ contains $a, b$ with $ab = 1 \\ne ba$.\n\nTaking $n = 1$ shows that a stably finite ring is Dedekind-finite.\nThe converse fails: there are Dedekind-finite rings $R$ for which $M_2(R)$ is not Dedekind-finite.\nEquivalently, $R$ is stably finite if for every $n \\ge 1$ the left module $R^n$ is not isomorphic to $R^n \\oplus N$ for a nonzero module $N$.\nA nonzero stably finite ring has invariant basis number.\n",
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