Unramified morphism of rings

Summary

An unramified morphism of rings is a map $R \to A$ of commutative rings that makes $A$ a finitely generated $R$-algebra. Moreover its module of Kähler differentials $\Omega_{A/R}$ must be zero. It generalizes finite separable field extensions and unramified extensions of rings of integers in number theory. Locally, take a prime $\mathfrak q$ of $A$ and the prime $\mathfrak p$ of $R$ below it. Then $\mathfrak p$ generates the maximal ideal of $A_{\mathfrak q}$, and the residue field extension is finite and separable. An unramified map that is also flat and finitely presented is étale.