TITLE: $\mathbb Q+X\mathbb R[X]$ is not Noetherian
QUESTION [6 upvotes]: Let $A=\{q+r_1X+ \cdots +r_nX^n: q \in \mathbb{Q}, r_i \in \mathbb{R}\}$ be the polynomial ring with rational costant terms. I have to prove that $A$ isn't a noetherian ring. How can I prove it?

REPLY [6 votes]: consider the chain of ideals:
$\pi xA \subset \pi xA + \pi^2xA \subset\pi xA +\pi^2 xA +\pi^3 xA \subset ...$