Let $B$ be a noetherian integral domain, let $A$ be a subring of $B$ such that $B$ is a finitely generated $A$ algebra. Assume that $A$ is also noetherian. Let $b$ be a non-zero element of $B$. How to show that there exists a non-zero element $a \in A$ such that for every algebraically closed field $K$, and every morphism$ f : A \to K$ such that $f(a)$ is not equal to $0$ can be extended to a morphism $g : B \to K$ such that $g(b)$ is not equal to $0$.
This is an exercise in Hartshorne Chapter 2 section 3. Problem number 3.19. I did the case when $B=A[X]$ a polynomial ring. How to do the case when $B=A[c]$ where $c \in B$. Thank you for any help.