Corollary 3

Let A be an integral domain, B a finitely generated A-algebra and b an element of B such that $zb^n \neq 0$ for all $z \neq 0$ in A and every integer $n > 0$. Let $\varphi : A \to B$ be the canonical homomorphism; there exists $a \neq 0$ in A such that, for every homomorphism f from A to an algebraically closed field L such that $f(a) \neq 0$, there exists a homomorphism g from B to L such that $g(b) \neq 0$ and $f = g \circ \varphi$.

The hypothesis on b implies that, if h is the canonical homomorphism $x \mapsto x/1$ of B to $B[b^{-1}]$, the homomorphism $h \circ \varphi$ of A to $B[b^{-1}]$ is injective. By Corollary 1 there therefore exist an element $a \neq 0$ of A and a subring $B'$ of $B[b^{-1}]$ such that $B[b^{-1}, a^{-1}]$ is integral over $B'[a^{-1}]$ and $B'$ is isomorphic to a polynomial algebra $A[Y_1, \ldots, Y_n]$. Let f be a homomorphism from A to an algebraically closed field L such that $f(a) \neq 0$; there exists a homomorphism from $A[Y_1, \ldots, Y_n]$ to L extending f and hence there exists a homomorphism $f'$ from $B'$ to L extending f. As $f'(a) \neq 0$ in L, there exists a homomorphism $f''$ from $B'[a^{-1}]$ to L such that

for all $x \in B'$ and all $n > 0$ (Chapter II, §2, no. 1, Proposition 1). Finally, as $B[b^{-1}, a^{-1}]$ is integral over $B'[a^{-1}]$, there exists a homomorphism $f'''$ from $B[b^{-1}, a^{-1}]$ to L extending $f''$ (§2, no. 1, Corollary 4 to Theorem 1). If $j : x \mapsto x/1$ is the canonical homomorphism from B to $B[b^{-1}, a^{-1}]$, $g = f''' \circ j$ solves the problem for $j(b)$ is invertible in $B[b^{-1}, a^{-1}]$ and hence $f'''(j(b)) \neq 0$ in L.

Note that, if B is assumed to be an integral domain and $A \subset B$ in Corollary 3, the hypothesis on b is equivalent to $b \neq 0$.