The question is the following: let A be a ring (commutative, with 1) whose additive group is a finitely generated abelian group. If $P\subset A$ is a maximal ideal, then $A/P$ is a finite field? To show this, I've argued as follows, but I'm not sure that this is the "right" way: A has a structure of finitely generated $\mathbb{Z}$-algebra. If $p\mathbb{Z} = P \cap \mathbb{Z}$, then, as a Corollary of a generalized form of Hilbert's Nullstellensatz, $A/P$ is a finite field extension of $\mathbb{Z}_p$, hence a finite field.
Is this a correct proof? thanks again!
