Idempotent ideal
= Idempotent ideal
{title2=$I^2=I$}
An <ideal> $I$ is idempotent when $I^2=I$. Then $I^j=I$ for every $j\ge1$, so a nonzero idempotent ideal gives a nonzero intersection of all its powers. A <semigroup algebra> with arbitrarily divisible positive exponents supplies examples in a non-Noetherian <integral domain>. If $I$ is finitely generated, the <determinant trick> applied to $I=I^2$ produces $r\in I$ such that $(1+r)I=0$.