Prove that the ring $\mathbb{Q}[x,y,z]\over (x^2,y^3,z^7)$ is Artin.

Question

I was trying to show that given a maximal ideal, every decreasing chain terminates. So, I found a maximal ideal $(x,y,z)$ in $\mathbb{Q}[x,y,z]\over (x^2,y^3,z^7)$ and proved it is maximal. Also, any decreasing chain from $(x,y,z)$ should terminate. But, I have no idea with how to prove it for the whole maximal ideals in $\mathbb{Q}[x,y,z]\over (x^2,y^3,z^7)$. I think there must be a nice way.

Answer 1

The ring $\Bbb Q[x,y,z]/(x^a,y^b,z^c)$ is finite dimensional over as a vector space $\Bbb Q$, so every descending chain of vector subspaces stabilises (in particular every descending chain of ideals stabilises). As a vector space it has dimension $abc$ and a basis consists of $x^i y^j z^k$ where $0\le i<a$, $0\le j<b$, $0\le k<c$.

Answer 2

$\{x^iy^jz^k\mid 0\leq i\leq 1, 0\leq j\leq 2,0\leq k\leq 6\}$ is a $\mathbb Q$ basis, so it is a finite dimensional $\mathbb Q$ vector space, which is of course, Artinian.