TITLE: Finitely generated ring with zero Krull dimension
QUESTION [5 upvotes]: I'm trying to prove the following:

Every finitely generated ring with Krull dimension equal to zero is finite.

I'm trying to show that the ring is a domain, hence a field, in order to use the property that   claims that every finitely generated field is finite, but I'm not seeing how I can show that this ring is a domain, so I guess that this way may, may be not the best way to show this.
Thank you for any help!

REPLY [8 votes]: To see this, note that since $R$ is a finitely generated $\mathbb{Z}$-algebra, that it is necessarily Noetherian. Thus, $R$, being Noetherian and dimension zero, it is necessarily Artinian. Thus, we have a decomposition of $R$ as $R\cong R_1\times\cdots\times R_n$ where each $R_i$ is a local Artinian ring, which is necessarily also finitely generated as a $\mathbb{Z}$-algebra. 
Now, it suffices to show that each $(R_i,\mathfrak{m}_i)$ is finite. It is a common fact that a local Artinian ring with finite residue field is necessarily finite itself. But, note that $R_i/\mathfrak{m}_i$ is a field which is finitely generated as a $\mathbb{Z}$-algebra, and so necessarily finite (this follows by the Nullstellansatz).