 Pure ideal

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.