Is the ring $\mathbb{C}[x,y]/(x^2,y^3)$ Artinian? If not, what is its Krull dimension?

Question

So, I know that the ring $\mathbb{C}[x,y]/(x^2,y^3)$ is Noetherian, since the ring $\mathbb{C}[x,y]$ is Noetherian.
In order to prove that the ring is not Artinian, I've tried finding a prime ideal that isn't also a maximal ideal, but I'm stuck. I believe that the ring is Artinian after all.
Any help would be greatly appreciated.

Answer 1

The ring is Artinian, indeed. The only prime ideal of $\mathbb C[x,y]$ which contains $(x^2,y^3)$ is $(x,y)$, so there is no prime non-maximal ideal in the factor ring.

Answer 2

Another argument why it is Artinian: Note that as a $\Bbb C$-vector space the ring is generated by $1,x,xy,xy^2,y,y^2$. This is because all higher powers of $x,y$ become zero in the quotient. So $R=\Bbb C[x,y](x^2,y^3)$ is finite dimensional as a $\Bbb C$-vector space. Every ideal of $R$ is in particular a subspace, hence any decreasing chain of ideals will stabilize because the dimension is finite, so $R$ is Artinian (this is a special case of the theorem that any f.g. module over an Artinian ring is also Artinian.)