Example of a noetherian domain which is neither Dedekind domain nor UFD?

Question

What is an example of a noetherian domain which is not Dedekind domain as well as not a UFD. I don’t have any clue with this. Also, we know that a noetherian domain is a factorisation domain! But is the converse true or are there any example of a factorisation domain which is not noetherian.

Thanks in advance for any kind of help!

Answer 1

The DaRT query suggested a result:

$k[[x^2,x^3]]$

I am unpracticed with these conditions, but here's what I think: it's not a UFD because $x^6$ factors in two ways, and I think it's not Dedekind because $(x^6)$ factors in two ways.

$k[X_i\mid i\in\mathbb N]$ is a UFD that's not Noetherian. (I gave this earlier, then removed it, but I'd like to put it back.)

For a different example that isn't a UFD either, Anne Grams produced an example of a non-Noetherian atomic domain (DaRT query), which should be adequately cited there.

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Update: I've updated the answer to reflect that a previously reported result ($\mathbb Q[x,y]_{(x,y)}$) was invalid. Owing to a typo in a particular entry, it was misclassified. It's been fixed for all affected rings. I owe a big debt of gratitude for uncovering this problem so that I could correct it. Thanks)

Answer 2

$k[x,y,z,w]/(xy-zw)$ works. It takes a bit to check.

Answer 3

If $k$ is a field the ring $k[X_1,...,X_n,...]$ of polynomials in denumerably many indeterminates over $k$ is a non-noetherian factorization domain (even a UFD actually).

Answer 4

A simple way to get infinitely many examples: if $A$ is a Dedekind domain with nontrivial class group then $A[x_1,\ldots,x_n]$ is an example for all $n \geq 1$. For instance, $\mathbf Z[\sqrt{-5}][x_1,\ldots,x_n]$ fits your conditions. The explanation below may involve things you have not seen, but at least the examples are easily appreciated.

If $A$ is Noetherian then so is $A[x]$, and $\dim A[x] = \dim A + 1$ (Krull dimension). Since Dedekind domains are $1$-dimensional, if $A$ is a Dedekind domain then $A[x]$ is not Dedekind but it is still Noetherian, and the same is true of $A[x_1,\ldots,x_n]$ for all $n \geq 1$.

Do you know what a Krull domain is? Dedekind domains are the $1$-dimensional Krull domains, and if $A$ is a Krull domain then so is $A[x]$. Each Krull domain has a class group, UFDs are the Krull domains with trivial class group, and if $A$ is a Krull domain then $A[x]$ has the same class group as $A$; that generalizes the fact that if $A$ is a UFD then so is $A[x]$. Therefore if $A$ is a Krull domain that is not a UFD then so is $A[x]$, and the same is true of $A[x_1,\ldots,x_n]$ for all $n\geq 1$.

From the previous paragraphs, if $A$ is Dedekind then $A[x_1,\ldots, x_n]$ is Noetherian, has dimension $n+1$, and has the same class group as $A$, so when $A$ has a nontrivial class group then $A[x_1,\ldots,x_n]$ fits your conditions for all $n\geq 1$. This provides infinitely many examples from one example of a Dedekind domain that is not a UFD.

Answer 5

I'm a bit late to the party, but here are a couple of examples that haven't been mentioned in the previous answers.

First example. The ring $R=\mathbf{Z}[\sqrt{-3}] = \mathbf{Z}[X]/(X^2+3)$ is a simple example of a Noetherian domain which is not integrally closed (since the element $\frac12 + \frac12 \sqrt{-3}$, which lies in $R$'s field of fractions $\mathbf{Q}(\sqrt{-3})$ but not in $R$, is a root of a monic polynomial in $R[X]$, namely $X^2 - X + 1$). Since Dedekind domains and UFDs are known to always be integrally closed, $R$ is therefore not a Dedekind domain and not a UFD.

Some background material for the second example:

• If $A$ is an integral domain with field of fractions $K$, let $\operatorname{Int}(A)$ denote the subring of $K[X]$ consisting of the polynomials which map $A$ into $A$. (So-called “integer-valued polynomials”, the name coming from the special case $A=\mathbf{Z}$.)
• A bounded factorization domain (BFD) is a factorization domain such that for each nonzero nonunit $x$ there is an $N$ such that all factorizations of $x$ have length at most $N$. Any UFD, such as $\mathbf{Z}$, is clearly a BFD.
• $\operatorname{Int}(A)$ is a BFD if and only if $A$ is a BFD. (This is Prop. VI.3.2 in the book Integer-Valued Polynomials by Cahen & Chabert.)

Second example. The ring $\operatorname{Int}(\mathbf{Z})$ is a BFD (since $\mathbf{Z}$ is a BFD), which is not Noetherian (Cahen & Chabert, Prop. V.2.7). In fact, it's a Prüfer domain (hence integrally closed) which is not a Dedekind domain (which implies that it's not a UFD either).

Similarly, the ring $\operatorname{Int}(\operatorname{Int}(\mathbf{Z}))$ is also a BFD (since $\operatorname{Int}(\mathbf{Z})$ is a BFD), which is not Noetherian. It's integrally closed but not a Prüfer domain; see this question.