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  "body": "\n## Definition\n\nLet $\\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$ and let $a \\in S$.\n\n1. The element $a$ is *left cancellable* if $a \\ast b = a \\ast c$ implies $b = c$, for all $b, c \\in S$.\n2. The element $a$ is *right cancellable* if $b \\ast a = c \\ast a$ implies $b = c$, for all $b, c \\in S$.\n3. The element $a$ is *cancellable* if it is both left and right cancellable.\n\nThe pair $(S, \\ast)$ is *left cancellative*, *right cancellative* or *cancellative* if every element of $S$ is left cancellable, right cancellable or cancellable.\n\n## Remarks\n\nExample: $(\\N, +)$ is cancellative, because $a + b = a + c$ gives $b = c$ by comparing the numbers, and $+$ is commutative.\n\nNon-example: $(\\Z, \\cdot)$ is not cancellative, because $0 \\cdot 1 = 0 \\cdot 2$ with $1 \\ne 2$, so $0$ is not left cancellable.\n\nAn element $a$ is left cancellable if and only if the map $x \\mapsto a \\ast x$ is injective.\nOn a set with at least two elements, the operation $a \\ast b = b$ is left cancellative but not right cancellative.\n",
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