# Integrally Closed (Normal) Domains

Being integrally closed is an important condition in commutative algebra. I'm trying to develop an understanding of this. So, I would like to know more about its motivation and intuitive significance.

There are many places I could go to learn about this, but the reason I want to ask here is because I have some specific smaller questions that I can't seem to find answers to. They are:

(1) Is there a way to think of integrally closed (normal) domains in terms of a universal property? The integral closure of a commutative ring $$R$$ should be the universal integrally closed domain which $$R$$ maps into. Put perhaps there is more to say here.

(2) Can integrally closed be expressed in terms of Galois theory, maybe? Take an extension of domains $$R \subset S$$. How does being integrally closed relate to the automorphism group $$\text{Aut}_R (S)$$?

(3) I understand that being normal is close to being smooth. Maybe this is related to saying, for a domain $$R$$, with $$K$$ its field of fractions, $$R$$ is normal if and only if $$R = \{ a \in K : d (a) = 0 \forall d \in \text{Der}_R (K, K) \}$$ where $$\text{Der}_R (K, K) = \{ d \in \text{Hom}_{R \text{-mod}} (K, K) : d(ab) = a d(b) + d(a) b \}$$ But I'm not actually sure about that. ($$\text{Der}_R(K, K)$$ is related to some sort of tangent space geometrically).

Your guess on (1) is correct. There may not be a relation as you ask for in (2). Even for field extensions, $$Aut_R(S)$$ could be trivial. (3) could be false in characteristic $$p>0$$.