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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nAn element $e \\in R$ is idempotent if $e^2 = e$.\n\n## Remarks\n\nIn every ring, $0$ and $1$ are idempotent.\nIn $\\mathbb{Z}/6\\mathbb{Z}$, the classes of $3$ and $4$ are idempotent, because $9 \\equiv 3$ and $16 \\equiv 4 \\pmod 6$.\nIf $e$ is idempotent, then $1 - e$ is idempotent, and $e (1 - e) = 0$.\nIn the [[product|0MNZCN4BQK]] $R \\times S$, the elements $(1, 0)$ and $(0, 1)$ are idempotent.\n\n## Explanations\n\n### L1 human\n\nAn idempotent does not change when you square it.\nAmong the integers, only $0$ and $1$ have this property.\nOther rings can have more idempotents, for example the projection matrices.\n",
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