Let $A\subseteq B$ be two noetherians domain with fraction fields $k$ and $L$, rispectively. Assume that $A$ is normal and $B$ is finite as $A$-module. I have to check that $B$ is also normal if $\Omega^1_{B/A}=0$. Can you help me?
I know that if $\Omega^1_{B/A}=0$ then $L/k$ is finite separable extentesion. Maybe is true that if $A\subset B$ is as above and $L/k$ is separable then $B$ is normal?
EDIT: By reading something about ring homomorphisms as above I found that the setup where I have to work is essentially that of unramiefied ring homomorphisms. Now I'm lonking for the simplest proof of the fact above.