# $\mathbb{R} ^ \mathbb{R}$ is a commutative ring with identity that is neither noetherian nor artinian.

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.

• Just do it. Where are you stuck? – Berci Jan 8 at 8:11
• I don't know how to show that it's neither noetherian nor atrinian. Don't have any idea about taking the ideals... @Berci – t.ysn Jan 8 at 8:14
• Does an approach similar to this one math.stackexchange.com/a/4051/42969 work? – Martin R Jan 8 at 8:18
• 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 $Z(A)\supsetneq Z(B)$. – egreg Jan 8 at 8:43
• Same strategy as this works. – 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.

• I get the distinct sense this website is slowly dying... – goblin Jan 8 at 9:44
• Thank you so much. I'm still a little bit confused but I try to understand it... – t.ysn Jan 8 at 12:58
• @t.ysn, which part is unclear? (I'll answer tomorrow since I'm going to bed now) – goblin Jan 8 at 14:15
• 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." – rschwieb Jan 8 at 15:05
• @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? – goblin Jan 9 at 8:49