Pure ideal
= Pure ideal
{title2=$A/I\text{ is flat over }A$}
An <ideal> $I$ is pure when the quotient is a <flat module>. Equivalently every $a\in I$ satisfies $a=ab$ for some $b\in I$. The latter condition makes $I$ vanish after localizing at primes containing it, while at primes not containing it the quotient is zero. Pure ideals need not be finitely generated. They describe <flat morphisms> that are <closed immersions> on affine charts.