let $R=\mathbb{R}^ \mathbb{R}$ (all the functions like $f:\mathbb{R} \rightarrow \mathbb{R}$). For each $f, g \in R$ and $a \in R$:

$$(f+g)(a):=f(a)+g(a)$$ $$(fg)(a):=f(a)g(a)$$

I want to show that $R$ is a commutative ring with identity which is neither noetherian nor artinian.

  • 4
    $\begingroup$ Just do it. Where are you stuck? $\endgroup$ – Berci Jan 8 at 8:11
  • $\begingroup$ I don't know how to show that it's neither noetherian nor atrinian. Don't have any idea about taking the ideals... @Berci $\endgroup$ – t.ysn Jan 8 at 8:14
  • 1
    $\begingroup$ Does an approach similar to this one math.stackexchange.com/a/4051/42969 work? $\endgroup$ – Martin R Jan 8 at 8:18
  • $\begingroup$ Hint: If $A$ is any subset of $\mathbb{R}$, then $Z(A)=\{f\in R:f(x)=0,\forall x\in A\}$ is an ideal of $R$; if $A\subsetneq B$, then $Z(A)\supsetneq Z(B)$. $\endgroup$ – egreg Jan 8 at 8:43
  • $\begingroup$ Same strategy as this works. $\endgroup$ – rschwieb Jan 8 at 14:58

Let's prove it's neither Noetherian nor Artinian.

To each $A \subseteq \mathbb{R}$, we can assign a set $\varphi(A) \subseteq \mathbb{R}^\mathbb{R}$ as follows:

$$\varphi(A) = \{f \in \mathbb{R}^\mathbb{R} : \forall x \in A, f(x) = 0\}.$$

Exercise 0. Show that $\varphi(A)$ is an ideal for all sets $A \subseteq \mathbb{R}$.

Conclude that $$\varphi : \mathcal{P}(\mathbb{R}) \rightarrow \mathcal{P}(\mathbb{R}^\mathbb{R})$$ can be viewed as a function $$\mathcal{P}(\mathbb{R}) \rightarrow \mathrm{Ideal}(\mathbb{R}^\mathbb{R}).$$

Exercise 1. Show that $\varphi : \mathcal{P}(\mathbb{R}) \rightarrow \mathrm{Ideal}(\mathbb{R}^\mathbb{R})$ is injective and order-reversing.

Exercise 2. Find an order-reversing injection $\psi : \mathbb{Z} \rightarrow \mathcal{P}(\mathbb{R})$.

Conclude that $\varphi \circ \psi : \mathbb{Z} \rightarrow \mathrm{Ideal}(\mathbb{R}^\mathbb{R})$ is an order-preserving injection.

Conclude that $\mathbb{R}^\mathbb{R}$ is neither Artinian nor Noetherian.

  • $\begingroup$ I get the distinct sense this website is slowly dying... $\endgroup$ – goblin Jan 8 at 9:44
  • $\begingroup$ Thank you so much. I'm still a little bit confused but I try to understand it... $\endgroup$ – t.ysn Jan 8 at 12:58
  • $\begingroup$ @t.ysn, which part is unclear? (I'll answer tomorrow since I'm going to bed now) $\endgroup$ – goblin Jan 8 at 14:15
  • $\begingroup$ This answer is interesting and OK... but it seems to be going out of its way to go over the posters head! I think a good compromise that keeps it a hint would be to add something like "consider what $\phi$ does to the chain of sets of the form $[-1/n, 1/n]$ and $[-n,n]$ for varying $n\in \mathbb N^+$ ordered by inclusion." $\endgroup$ – rschwieb Jan 8 at 15:05
  • $\begingroup$ @rschwieb, I take your point, but for me, such hints were always very confusing, perhaps because my reading comprehension skills aren't very strong. I try to offer people a bit more formality than is usual in "hint" answers for this reason. Anyway, why not post your own answer referencing the $\varphi$ function I've just defined and then reconceptualizing things in your own terms? $\endgroup$ – goblin Jan 9 at 8:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.