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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.

A morphism $f:X\to Y$ is separated when its diagonal $X\to X{\times }_{Y}X$ is a closed immersion. This is the scheme-theoretic analogue of the Hausdorff property.

A morphism of schemes is separated when its diagonal is a closed immersion. This is a property of the morphism, distinct from the broader morphism or diagonal constructions.