Example of a commutative ring that is Artinian but not Noetherian 
I want to give an example of a commutative ring that is Artinian but not Noetherian. Is there any examples not very difficult?

I considered the ring $\mathbb{Z}+x\mathbb{Q}[x]$. It is not Noetherian. But can we prove it is Artinian or not?

Consider the abelian group $R=\mathbb Z_{p^\infty }$. We define $xy=0$ for every $x,y\in R$. 
Now, it is not hard to see that $R$ is an artinian ring but $R$ is not a noetherian ring.
Note that there is a prominent theorem telling about the relationship between artinian rings and noetherian rings.

Akizuki–Hopkins–Levitzki theorem:
  Let $R$ be a unitary ring. If $R$ is a left (right) artinian ring, then  $R$ is automatically a left (right) noetherian ring.

