An example of a non Noetherian UFD

Question

An example of a non Noetherian UFD.

I know an example is $$K[x_1,\ldots,x_n,\dots]$$ with $K$ a field, but I don't know why. Can someone give another example or better an explanation?

Is it not Noetherian because it's not finitely generated? And why do you know it is a UFD?? My question now is how to prove this is a UFD.

Answer 1

A ring $R$ has a factorization if it's Noetherian. Of course the factorization must not be unique. For the unicity you have to assume that every irreducible is prime.

In your example, $K[x_1, ..]$ is a UFD since $K$ is UFD and each polynomial has a finite number of variables. Furthermore it's not Noetherian because $(x_1, x_2...)$ is not finitely generated, however $K[x_1, ..]$ is finitely generated since $(1) = K[x_1, ...]$

Answer 2

The essential point is that the polynomial ring in infinitely many variables is the ascending union of subrings $K[x_1,\ldots,x_n]$, since any polynomial can involve only finitely-many indeterminates. Each of these rings is a UFD, and it is easy to see that a polynomial in which $x_N$ does not appear has only factorizations in which $x_N$ does not appear, again because everything takes place inside some polynomial ring in finitely-many variables. But the ring is not Noetherian, because the ideal generated by all the indeterminates is certainly not finitely-generated.

Edit: in response to @rschwieb's comment/query about why $x_N$ cannot appear in any factorization of a polynomial $P(x_1,\ldots,x_n)$ not already involving it... If it did, then not only does $P$ have have its factorization in the UFD $K[x_1,\ldots,x_n]$, but also (allegedly) in the UFD $K[x_1,\ldots,x_n,\ldots,x_N]$, with $x_N$ in the latter, not in the former. Impossible.

Answer 3

A Noetherian ring will satisfy the Ascending Chain Condition, or equivalently, every proper ideal will be finitely generated. Your example is not Noetherian for both of these reasons, as we can see $$(x_1)\subseteq (x_1,x_2)\subseteq (x_1,x_2,x_3)\subseteq\cdots,$$ and the chain never stabilizes since each indeterminate is distinct from the others. Likewise, $$(x_1,x_2,\ldots)$$ can't be finitely generated since there is no way to generate $x_n$ from $(x_1,\ldots,x_{n-1},x_{n+1},\ldots)$.

Another example of a non-Noetherian ring, for example is $$\prod_{n=1}^\infty \Bbb Z/2\Bbb Z.$$ Can you see why based off of your example and my explanation?