Give an example of an integral domain which is noetherian but not dedekind
Answers
A simple way to get infinitely many examples: if A is a Dedekind domain with nontrivial class group then A[x1,…,xn] is an example for all n≥1. For instance, Z[−5−−−√][x1,…,xn] 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 dimA[x]=dimA+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[x1,…,xn] for all n≥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[x1,…,xn] for all n≥1.
From the previous paragraphs, if A is Dedekind then A[x1,…,xn] is Noetherian, has dimension n+1, and has the same class group as A, so when A has a nontrivial class group then A[x1,…,xn] fits your conditions for all n≥1. This provides infinitely many examples from one example of a Dedekind domain that is not a UFD.