 Scheme-theoretic image

The scheme-theoretic image of $f:X\to Y$ is the smallest closed subscheme of $Y$ through which $f$ factors. For an affine morphism $\mathrm{Spec}B\to \mathrm{Spec}A$ induced by $\phi :A\to B$, it is $\mathrm{Spec}(A/\ker \phi )$, whose underlying set is the closure of the set-theoretic image.

The construction retains the scheme structure on the closed subscheme, not merely the set of points in an image.