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  "body": "\n## Definition\n\nLet $(M, \\ast, e)$ be a [[monoid|0N6R902FG1]].\nAn element $u \\in M$ is a unit when it has a two-sided [[inverse|0ZNZH0EXFX]], that is, an element $v \\in M$ with $u \\ast v = v \\ast u = e$.\nThe group of units of $M$ is the set $M^\\times$ of the units of $M$, with the operation $\\ast$ restricted to $M^\\times$.\nIt is a [[group|G5S1M9]] with identity $e$.\n\n## Remarks\n\nThe units of the monoid $(\\mathbb{Z}, \\cdot)$ are $1$ and $-1$.\nThe units of the monoid of $n \\times n$ real matrices under multiplication are the invertible matrices, and they form the general linear group $\\mathrm{GL}_n(\\mathbb{R})$.\nFor a ring $R$, the group of units $R^\\times$ is the group of units of the monoid $(R, \\cdot)$.\n\n## Explanations\n\n### L1 human\n\nIn a monoid, some elements can be undone and others cannot.\nThe elements that can be undone are the units.\nThe units alone form a group, because products and inverses of units are units.\n",
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