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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with $1$.\nFor an integer $n \\ge 0$, let $R^n$ be the [[free left module|0DZ9DKAKZR]] of rank $n$ over $R$.\nThe ring $R$ has *invariant basis number* (IBN) if, for all integers $m, n \\ge 0$, an isomorphism $R^m \\cong R^n$ of left $R$-[[modules|0NZ8JA25WK]] implies $m = n$.\n\n## Remarks\n\nExample: every nonzero commutative ring $R$ has IBN.\nTo check it, take a maximal ideal $\\mathfrak{m}$ of $R$; an isomorphism $R^m \\cong R^n$ gives an isomorphism $(R/\\mathfrak{m})^m \\cong (R/\\mathfrak{m})^n$ of vector spaces over the field $R/\\mathfrak{m}$, so $m = n$ by dimension.\n\nNon-example: let $V$ be a vector space of countably infinite dimension over a field $k$ and let $R = \\End_k(V)$.\nSince $V \\cong V \\oplus V$, one gets $R \\cong R^2$ as left $R$-modules, so $R$ does not have IBN.\n\nThe zero ring does not have IBN, since $0 = R^m$ for all $m$.\nThe condition can be stated with matrices: $R$ has IBN if and only if there are no matrices $A \\in R^{m \\times n}$ and $B \\in R^{n \\times m}$ with $m \\ne n$, $AB = I_m$ and $BA = I_n$.\nThis form is symmetric in left and right, so the definition does not depend on the side.\n",
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