Give an example of ring that has two maximal chains of prime ideals of different lengths.

Question

Give an example of ring that has two maximal chains of prime ideals of different lengths.

Is there an easy example?

Answer 1

$\mathbb Z\times F$ where $F$ is a field.

One maximal chain is $\mathbb Z\times\{0\}$ (of length $0$)

and another is $\{0\}\times F\subseteq (p)\times F$ for a prime $p$.