Examples of Noetherian Domains of Dimension One

Question

I am writing an assignment on Noetherian domains of dimension 1. I have been researching online left and right looking for concrete examples of this domain, but could not find any. I would therefore appreciate you very much if you could help me with links, pointers or outright examples of this domain. I would like examples that are simple, not requiring higher-end tools, befitting the mind of a beginner.

Answer 1

Every finite ring extension of $\mathbb Z$ is a Noetherian ring of dimension one. For instance, $\mathbb Z[\sqrt{-3}]$ is a Noetherian domain of dimension one (which is not Dedekind).

Answer 2

All Dedekind domains are noetherian domains of dimension $1$. Rings of algebraic integers, in arithmetic, are Dedekind domains, some are principal. If they're UFD's they're principal.

Actually, one characterization of Dedekind domains is they're noetherian integrally closed domains of dimension $1$.

Answer 3

If $k$ is a field and $f(X,Y)\in k[X,Y]\setminus k$ is an irreducible polynomial, then the quotient ring $A=k[X,Y]/\langle f(X,Y)\rangle$ will be a noetherian domain of dimension one, not necessarily Dedekind.
For example if $k=\mathbb Q$ the ring $A=\mathbb Q[X,Y]/\langle X^n+Y^n-1\rangle$ is a Dedekind domain for all $n\geq 1$, but the one-dimensional noetherian domain $R=\mathbb Q[X,Y]/\langle X^3-Y^2\rangle$ is not Dedekind.

Geometric interpretation (optional)
The ring $A$ is the ring of regular functions $A=\mathcal O(V)$ of the affine plane curve $V\subset \mathbb A^2_k \;$ given by the equation $f(x,y)=0$.
The geometric point of view is very fruitful, for example if one wants to investigate whether $A$ is Dedekind.