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  "body": "\n## Definition\n\nLet $S$ be a set and let $\\ast$ be a [[binary operation|0M418FE7BB]] on $S$.\nAn element $e \\in S$ is a left identity when $e \\ast a = a$ for all $a \\in S$.\nAn element $e \\in S$ is a right identity when $a \\ast e = a$ for all $a \\in S$.\nAn element $e \\in S$ is an identity element (or two-sided identity) when it is both a left identity and a right identity.\n\n## Remarks\n\nThe number $0$ is the identity element for addition on $\\mathbb{Z}$, and $1$ is the identity element for multiplication.\nIf $e$ is a left identity and $f$ is a right identity, then $e = e \\ast f = f$.\nThus an operation has at most one two-sided identity element.\n\n## Explanations\n\n### L1 human\n\nAn identity element is an element that does nothing when you combine it with another element.\nAdding $0$ to a number gives the same number, and multiplying a number by $1$ gives the same number.\nSome operations have no identity element, for example addition on the positive integers.\n",
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