Noether polynomial ring over $2\Bbb Z$ 
Given a definition of $R=2\Bbb Z$ prove that ring $R[x]$ is not noether.

I assume that the proof should be based on the fact that a ring is noether if and only if any ideal is finitely generated. However, I'm struggling with constructing a non-finitely generated ideal, because only ideals I can think of are "root-based" (i.e. polynomials that have certain roots), that are, of cource, finitely generated.
I would be very grateful for a description of such ideal or a piece of work that can explain the structure of polynomial ring ideals in detail.

It's unconventional to treat rings without multiplicative identity, but here goes.
Let $R=2\Bbb Z$.
There is an easy example of a non-finitely generated $R[X]$-ideal,
namely $R[X]$ itself. To see an infinite ascending chain of $R[X]$-ideals,
let $I_n$ be the ideal generated by $2X$, $2X^2,\ldots,2X^n$.  
