I wonder if and where the separability condition on $L$ is used in the following theorem from Lang's Algebraic Number Theory p. 7. I suspect it is necessary to see that a subring of the finite extension $L$ is a finitely generated $A$-module, but have problems imagining a counterexample to this theorem without separability. Even if this exists, a lot of the inseparable extension still suffice the assertion.
Let me post a more elaborate version of the proof:
$B$ is a torsion-free $A$-module, because the multiplication $xa = 0$ for $x\in B$ and $a\in A$ does not allow zero-divisors as it happens in $L$. (Now we would need to deduce that $B$ is finitely generated over $A$.) From the theory of PIDs $B$ is free. Assuming $B$'s dimension is smaller than $n$, and its basis shall be $\alpha_1,\dots,\alpha_k$. Then we can find a $\beta \in L$ that is linearly independent from this basis and a $c\in A-\{0\}$ such that $c\beta$ is integral over $A$ but still $c\beta\in B$ is linearly independent from the basis of $B$. Contradiction.