= Idempotent ideal in a finite category
{title2=$I^2=I$}
A collection of arrows is a two-sided ideal when arbitrary precomposition and postcomposition preserve it. If a finite <category> has split <idempotent morphisms>, every ideal $I$ satisfying $I^2=I$ is generated by identities belonging to $I$. Factor an arrow $f\in I$ repeatedly as a longer product of arrows of $I$. Two prefixes coincide, since there are only finitely many arrows. The intervening endomorphism $t\in I$ fixes that prefix; an idempotent power $e$ of $t$ still fixes it. Write $e=ir$, $ri=1_B$. Then $1_B=rei\in I$, and $f$ factors through $B$. Applied to the least covering sieves of a finite site, this proves <rigid coverage>.
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