Let $A\subset B$, $C\subset D$ be commutative rings with identity and let $h:A\to C$ be an injective ring homomorphism. Can we always extend $h$ to a ring homomorphism $h’:\overline{A}\to \overline{C}$ where $\overline{A}$ is the integral closure of $A$ in $B$ and $\overline{C}$ the integral closure of $C$ in $D$?
I came up with this question when trying to prove chapter 5, exercise 2 in A&M, which states that for algebraically closed field $k$ we can always extend ring homomorphism $h: A\to k$ to $h:\overline{A}\to k$, where $\overline{A}$ is the integral closure of $A$ in a ring $B$ that contains it.
The answer to your question, as it is currently stated, is "no".
Simply take $A = \mathbb{Z} = C = D$ and $B = \mathbb{Z}[i]$.
The key point in the result you are citing is that $k$ is assumed to be algebraically closed. Then it is true that for any subring $A$ of $k$ and any algebraic extension $\overline{A}$ of $A$, one may embed $\overline{A}$ into $k$.
Behind this is an important lemma saying that if $A \subseteq B \subseteq C$ are ring extensions such that $B$ is integral over $A$ and $C$ is integer over $B$, then $C$ is integral over $A$.