= Solution
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 $R=\mathbb Q[x_1,x_2,\ldots]$, the <polynomial ring> in countably many variables, each individual <polynomial> using only finitely many variables. The ascending chain of <ideals>
$$
(x_1)\subsetneq(x_1,x_2)\subsetneq(x_1,x_2,x_3)\subsetneq\cdots
$$
is strictly increasing. Indeed, if $x_{n+1}$ belonged to $(x_1,\ldots,x_n)$, 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 \b[$\mathbb Q$ is Noetherian, while $\mathbb Q[x_1,x_2,\ldots]$ is not].
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