Ring homomorphism and affine scheme

Question

How to describe all ring homomorphisms $f: A \rightarrow B$, such that corresponding affine scheme morphism $f: Spec \, B \rightarrow Spec \, A$ is open immersion?

Answer 1

The answers in the link given by Manny are great. Let me just add another sufficient condition which may be simpler to check than, e.g. flatness. If

• $A$ is an integrally closed domain,
• $B$ is contained in $\mathrm{Frac}(A)$ and finitely presented over $A$ (as $A$-algebra),
• $f$ is quasi-finite (i.e. for all prime ideals $p$ of $A$, $B/pB$ is artinian),

then $f$ is an open immersion. This is a form of Zariski's Main Theorem.