What is an easy example of non-Noetherian domain?

Keep in mind, I'm strictly an amateur, though a very old one. I learned about imaginary numbers barely two years ago and ideals a year ago, and I'm still decidedly a novice in both topics.

In the university library, I was looking at Modules over Non-Noetherian Domains by Fuchs and Salce and I couldn't really understand anything. I'm also looking at the "Questions that may already have your answer," but if they do, it's not in a way that I can understand.

Then I thought what about a finite ring, like maybe $\mathbb{Z}_{10}$, but that's only created more questions, like: can a finite ring be non-Noetherian? Although $5 = 5^n$ for any $n \in \mathbb{Z}_{10}$ besides $0$, we're still dealing with only the ideal $\langle 5 \rangle$, right? There's no ascending chain of ideals even though some numbers in this domain have infinitely many factorizations, right? It is a Noetherian ring after all, right?

My question, it seems, has then become if it's possible for a non-Noetherian ring to be within the comprehension of a dilettante such as myself, or must it necessarily be esoteric and exotic?

Solution 1:

For a field $k$ the ring $k[x_1,x_2,\dots]$ of polynomials with infinite indeterminates is non-Noetherian because you can take the ascending chain

$$(x_1)\subset (x_1,x_2)\subset(x_1,x_2,x_3)\subset\cdots$$

And also to answer one of your questions, a finite ring must be Noetherian because an equivalent definition of Noetherian is "every ideal is finitely generated", so if $R$ is finite then each ideal $I$ is finite and in particular it's generated by itself, so every ideal is finitely generated.

Solution 2:

Being "Noetherian" can be read as a ring for which any ascending chain of ideals has a "biggest ideal", one that contains all the others but is only contained by ideals which are equal to itself.
To be non-Noetherian, the ring simply needs to have an infinite ascending chain of ideals. The ring of algebraic integers, for example, has the infinite chain of ideals generated by $2^{1/{2^{n}}}$.
That is, $$\langle \sqrt{2} \rangle \subset \langle \sqrt[4]{2} \rangle \subset \langle \sqrt[8]{2} \rangle \subset \langle \sqrt[16]{2} \rangle \subset \dots$$ forms a chain without a "biggest link".

Solution 3:

The ring of integer-valued polynomials (the subring of $\mathbf Q[x]$ of polynomials which take integer values at integers) is another example of a non-noetherian integral domain.

The ring of continuous functions on $[a, b]$ is yet another example (it's not an integral domain).