Let $(A,\mathfrak{p})\subset(B,\mathfrak{q})$ two local rings, such that $B$ is finitely generated as $A$-module. It's very well known, from Algebraic Geometry, that if $\Omega^1_{B/A}=0$ then the inclusion is unramified, hence in particular we must have $\mathfrak{q}=\mathfrak{p}B$. However I think that at last the equality $\mathfrak{q}=\mathfrak{p}B$ can be proved by using only ring theory, without schemes and such.

Any idea?