How to show that $k[x]$, polynomial ring over $k$ where $k$ is a field, does not satisfy the Descending Chain Condition.
The Descending Chain Condition is if $A_1 \supseteq A_2\supseteq A_3....... $ is a descending chain then it must become stationary i.e there is an $n$ such that $A_n = A_{n+k}$ $ \forall k\geq1 $
The polynomial ring $K[X]$, where $K$ is a field is a classical example that shows that Hilbert's basis theorem is no longer true if one replaces a noetherian ring for an artinian ring.
Indeed, every field $K$ is trivially both artinian and noetherian. Now, to show that $K[X]$ is not artinian, it's enough to find a infinite descending chain of ideals of $K[X]$. As the user Henno Brandsma points out in his comment, a good candidate is the family of ideals $\{A_n = \langle x^n\rangle: n\in \Bbb N\}$.
To prove the above claim we note that if such chain becomes stationary, then there would be some $k\in \Bbb N$ such that $\langle x^k\rangle = \langle x^{k+1}\rangle = \langle x^{k+2}\rangle =\ldots$
In particular the above implies that $x^{k+1}\mid x^k$, which is clearly absurd. Hence, we're done.