Examples of 1 dimensional Noetherian rings that aren't domains

Question

What are some examples of Noetherian Rings of Krull Dimension 1 that are not domains? It is relatively easy to find examples of domains(eg. $\mathbb{Z},\mathbb{F}[x]$) however I cannot seem to think of examples that are not domains.

Answer 1

The product of two Noetherian rings is Noetherian, and $\dim(A\times B)=\max\{\dim A,\dim B\}$. Therefore, $\Bbb Z\times \Bbb Z$ is like you requested.

Answer 2

Well, if a commutative ring $R$ is Noetherian and $I$ is an ideal of $R$, then $R/I$ is Noetherian.

$\Bbb Z$ is Noetherian but $\Bbb Z/\langle 6\rangle$ is not a domain as it contains zero-divisors.

Answer 3

Two more examples:

• $\mathbf Z/n\mathbf Z[X]$, where $n$ is composite.
• $K[X,Y]/(XY)$, where $K$ is a field.