Prove that the ring of continuous functions $f:\mathbb R\to\mathbb R$ is not Noetherian.
I know that to be Noetherian, every ideal is generated by finitely many elements or equivalently R satisfies the ascending chain condition.
So, if I can find an ideals that are contained in each other that don't terminate then it is not Noetherian.
My professor briefly touched on Noetherian rings so it is still a little bit confusing. How do I go about finding these ideals? Or should I show that every ideal is generated by finitely generated elements? Any help is much appreciated!
Best Answer
Consider $I_n=\{f \in C(\mathbb{R}):f(x)=0, \forall x \geq n\}$ ($n >0$). Then clearly $I_n$'s form an infinite ascending chain of ideals.