Corollary 3

Let $A, A'$ be two rings and $h$ a homomorphism from $A$ to $A'$. Suppose that $A$ is semi-local and Noetherian and that $A'$ is a finitely generated $A$-module. Then the ring $A'$ is semi-local and Noetherian; if $m$ is a defining ideal of $A$, $mA'$ is a defining ideal of $A'$.

We know that $A'$ is a Zariski ring with the $mA'$-adic topology (Chapter III, § 3, no. 3, Proposition 7). As $A/m$ is Artinian (Corollary 2) and $A'/mA'$ is a finitely generated $(A/m)$-module, $A'/mA'$ is an Artinian ring, hence $A'$ is semi-local and $mA'$ is a defining ideal of $A'$ (Corollary 2).