Restricted to affine schemes, this yields, dually, a notion of étale morphisms between rings. Étale maps between noncommutative rings have also been considered.
Between analytic spaces
Étale maps between analytic spaces are closely related to étale maps between schemes, while also a special case of an étale map between smooth spaces.
Zoran: I do not understand this statement. Analytic spaces have a different structure sheaf; in general nilpotent elements are allowed. This is additional structure not present in theory of smooth spaces.