Example of non-Noetherian ring

Question

Can you give an example of a non-noetherian ring $R[x_1,x_2]$ with an ideal which is not finitely generated?

Answer 1

Assuming $x_1$ and $x_2$ are indeterminates and $R$ is commutative, Hilbert’s basis theorem says that if $R$ is noetherian, then also $R[x_1,x_2]$ is noetherian. Hence you need $R$ non-noetherian.

The simplest example is then $R=k[x_3,x_4,\dots,x_n,\dotsc]$, the ring in infinitely many variables (countably many) over the field $k$. Then $R[x_1,x_2]$ is the same as $k[x_1,x_2,x_3,\dotsc]$ and a nonfinitely generated ideal is easy to find.