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Let $f: X \rightarrow Y$ be a morphism between schemes. When is the image of a closed point closed? In another question , some remarks were already made. For example if $X$ and $Y$ are of finite type over a field, then the image of a closed point is always closed (even if the map $f$ is not closed). On the other hand, examples like $Spec\, \mathbb{Q} \to Spec \, \mathbb{Z}_{(p)}$ provide us with examples of (flat, finite type) maps such that the image of a closed points is not closed.

Is there any reasonably general algebraic criterion when a morphism sends closed points to closed points if the schemes are not over a field?

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An integral morphism preserves (and also reflects) closed points. This is because for an integral extension of integral domains $D \subseteq D'$ we know that $D$ is a field iff $D'$ is a field.

Besides, there are many important and well-known strengthenings:

projective $\Longrightarrow$ proper $\Rightarrow$ closed (=preserves closed subsets) $\Rightarrow$ preserves closed points

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