Let $K$ be a field, $W$ an algebraic variety over $K$ and $A(W)=\{f:W\rightarrow K \mid f \text{ regular}\}$ the ring of regular functions. When is $A(W)$ an Artinian ring?
I know that $A(W)$ is a reduced finitely generated $K$-algebra, and I have tried to stabilize a descending chain $I_1\supseteq I_2 \supseteq \dots$ but I have not even fount a sufficient condition.
Any help is appreciated!