Let $A$ be a finitely generated $k$-algebra with no nilpotents. What do I need to show in order to prove it's a finite dimensional vector space over $k$?
For example, is it enough to show that $A$ has only finitely prime ideals and every prime ideal is maximal?
A finitely generated commutative $k$-algebra $A$ is finite dimensional as a $k$-vector space exactly when it is artinian.
Proof: finite-dimensional implies artinian is essentially trivial (each ideal is also a vector subspace, and there are finitely many of them). For the other direction, pick generators for $A$, ie a surjective homomorphism $p:k[x_1,\cdots,x_n]\to A$. Then as $A$ is artinian, $p(\prod x_i^{n_i})=p(\prod x_i^{n_i+1})=0$ for some collection of $n_i$. But then we have a composition series for $A$ with all sections finite dimensional vector spaces, so $A$ is a finite dimensional vector space. $\blacksquare$
So any condition implying a ring is artinian will work. So for example requiring every prime ideal to be maximal will work.
