# Hartshorne Exercise II. 3.19 (b)

How do we prove the following exercise of Hartshorne?

Let $A$ be a subring of an integral domain $B$. Suppose $B$ is a finitely generated $A$-algebra. Let $b$ be a non-zero element of $B$. Then there exists a non-zero element $a$ of $A$ with the following property. If $\psi\colon A \rightarrow \Omega$ is any homomorphism of $A$ to an algebraically closed field $\Omega$ such that $\psi(a) \neq 0$, then $\psi$ extends to a homomorphism $\phi\colon B \rightarrow \Omega$ such that $\phi(b) \neq 0$.

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Check Proposition $5.23$ in Atiyah-Macdonald.