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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).

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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 $p>0$.

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