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  "body": "\n## Definition\n\nLet $S$ be a set with a [[binary operation|0M418FE7BB]] $\\ast$ and an [[identity element|07WYQPB8E3]] $e$.\nLet $a, b \\in S$.\nThe element $b$ is a left inverse of $a$ when $b \\ast a = e$.\nThe element $b$ is a right inverse of $a$ when $a \\ast b = e$.\nThe element $b$ is an inverse of $a$ (or two-sided inverse) when it is both a left inverse and a right inverse of $a$.\nAn element that has an inverse is called invertible.\n\n## Remarks\n\nFor addition on $\\mathbb{Z}$, the inverse of $a$ is $-a$.\nFor multiplication on $\\mathbb{Q}$, each nonzero $a$ has the inverse $1/a$, and $0$ has no inverse.\nWhen $\\ast$ is [[associative|0K8F7PVDMX]], an element has at most one inverse.\n\n## Explanations\n\n### L1 human\n\nAn inverse of an element is a second element that cancels the first one.\nIf you add $5$ and then add $-5$, you come back to $0$, the identity for addition.\nFor multiplication, the inverse of $2$ is $\\tfrac{1}{2}$, because $2 \\cdot \\tfrac{1}{2} = 1$.\n",
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