Corollary 2

Let $A'$ be an integral domain, $A$ a subring of $A'$ such that $A'$ is integral over $A$ and $f$ a homomorphism from $A'$ to a ring $B$. If the restriction off to $A$ is injective $f$ is injective.

If $a'$ is the kernel off, the hypothesis means that $a' \cap A = (0)$; as $A'$ is an integral domain, Corollary 1 may be applied taking $p$ and $p'$ to be the ideal (0) of $A$ and the ideal (0) of $A'$ respectively, whence $a' = (0)$.