Proving the ring $\mathbb{Q}$[$\mathbb{Z}$] is not artinian

Question

My proposed solution:

For each $n \in \mathbb{N}$, $\mathbb{Q}$[$2^{n}\mathbb{Z}$] is an ideal of $\mathbb{Q}$[$\mathbb{Z}$] (I think) and so we have the following infinite descending chain of ideals:

$\mathbb{Q}$[$2\mathbb{Z}$] $\supsetneq$ $\mathbb{Q}$[$4\mathbb{Z}$] ... $\supsetneq$ $\mathbb{Q}$[$2^{n}\mathbb{Z}$] $\supsetneq$ ...

So the result follows.

Is this ok? Thanks.

Answer 1

Show $\Bbb Q[\Bbb Z]$ is isomorphic to the Laurent polynomials $\Bbb Q[x,x^{-1}]$ via an obvious map.

That ring is a domain. But an Artinian domain is a field. Is our ring a field?