TITLE: Nice exercises on Hilbert's basis theorem
QUESTION [8 upvotes]: I am teaching an extremely basic course in algebraic geometry and I would like to find more exercises on Hilbert's basis theorem, a result from commutative algebra. Unfortunately, I have not been able so far to find interesting exercises in all the books and lecture notes that I've seen.
Could you advise me some exercises on this topic?

REPLY [4 votes]: Here's an exercise from Ravi Vakil. The game of Chomp starts with a half-infinite checkboard with squares labeled $(n, m)$ for integers $n, m\geq 0$, with a cookie on each square. Two players take turns choosing a cookie (i.e., not an empty square) and eating all the cookies on that square and the ones that are not to the left or below it; that is, if a player chooses the cookie on $(n, m)$, then they eat all the cookies on $(n', m')$ for $n' \geq n$ and $m'\geq m$. The cookie on $(0, 0)$ is poisoned, and the player who eats it immediately loses. Prove that any game of Chomp ends after finitely many moves.
For the connection with the Hilbert basis theorem, consider $(n, m)$ as representing the monomial $X^n Y^m\in R[X, Y]$. More generally, consider the analogous game on $(\mathbb{Z}^{\geq 0})^d$.