Practice problem

Commutative Algebra

Recall that a ring $A$ is said to be a $k$ -algebra if it is equipped with a morphism of rings $k\to A$ . Let $k'$ be the the integral closure of $k$ in $A$ : the set of elements of $A$ integral over $k$ . Let $A^\times$ be the unit group of $A$ .

Let $k$ be a field and $A$ a $k$ -algebra. Let $k'$ be the integral closure of $k$ in $A$ . Suppose $A$ is an integral domain. Must $k'$ be a subset of $\{0\}\cup A^\times$ ?