Closed immersion

Summary

A closed immersion is a morphism of schemes that identifies its source with a closed subscheme of its target. For affine schemes, closed immersions into $\operatorname{Spec} A$ are exactly the maps $\operatorname{Spec} A/I \to \operatorname{Spec} A$ for ideals $I$. They make precise the idea of a subvariety cut out by equations, and they include non-reduced subschemes such as a double point.