Let $B$ be an integral ring extension of $A$ and $B$ is an integral domain. Let $f:B \to C$ be a ring homomorphism to any commutative ring $C$ such that the restriction of $f$ to $A$ is injective. Then I have to show that $f:B \to C$ is also injective.
It is clear from the diagram that $\ker(f) \cap A= (0).$ How can I show that $\ker(f)=(0)$ ? Help me.