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  "body": "\n## Definition\n\nLet $\\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$ and let $z \\in S$.\n\n1. The element $z$ is *left absorbing* if $z \\ast a = z$ for all $a \\in S$.\n2. The element $z$ is *right absorbing* if $a \\ast z = z$ for all $a \\in S$.\n3. The element $z$ is *absorbing*, or a *zero element*, if it is both left and right absorbing.\n\n## Remarks\n\nExample: in $(\\Z, \\cdot)$ the integer $0$ is absorbing, because $0 \\cdot a = a \\cdot 0 = 0$ for all $a \\in \\Z$.\n\nNon-example: in $(\\Z, +)$ the integer $0$ is not absorbing, because $0 + 1 = 1 \\ne 0$; it is an identity element instead.\n\nIf $z$ is left absorbing and $z'$ is right absorbing, then $z = z \\ast z' = z'$.\nSo an operation has at most one absorbing element.\nOn a set with at least two elements, the operation $a \\ast b = a$ makes every element left absorbing and no element right absorbing.\n",
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