Closed immersion

A morphism $i:Z\to X$ is a closed immersion when it identifies $Z$ homeomorphically with a closed subset of $X$ and the morphism ${\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_Z$ is surjective. Affine-locally it has the form $\mathrm{Spec}(A/I)\to \mathrm{Spec}A$.

A closed subscheme of $X$ is a scheme $Z$ together with a closed immersion $Z\hookrightarrow X$, usually identified with its image and its quotient structure sheaf.

Every closed subset $Z\subseteq X$ has a canonical reduced closed-subscheme structure defined affine-locally by $\mathrm{Spec}(A/\sqrt{I})$ when $Z=V(I)\subseteq \mathrm{Spec}A$. It is the smallest closed subscheme with underlying set $Z$.

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.