Let $f\colon (A,\mathfrak{p})\longrightarrow (B,\mathfrak{q})$ be a homomorphism of local rings. I have to check that if $\Omega^1_{B/A}=0$ then $\mathfrak{q}=\mathfrak{p}B$.
REMARK: I'm not sure if to prove the claim we need that $B$ is finitely generated over $A$ and if we need the injectivity of $f$.