Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 2 / 11G / i

A field, for example $\mathbb{Q}$, is a Noetherian ring: its only ideals are $(0)$ and the whole field, both finitely generated.

For a non-Noetherian example, take $R=\mathbb{Q}[x_1,x_2,\dots ]$, the polynomial ring in countably many variables, each individual polynomial using only finitely many variables. The ascending chain of idealsis strictly increasing. Indeed, if $x_{n+1}$ belonged to $(x_1,\dots ,x_n)$, setting the first $n$ variables to zero would give the impossible equality $x_{n+1}=0$. The chain never stabilizes, contradicting the ascending chain condition. Thus $\mathbb{Q}$ is Noetherian, while $\mathbb{Q}[x_1,x_2,\dots ]$ is not.