Integral domains examples

Question

I am supposed to give an example of
1) an infinite integral domain of characteristic $5$, and
2) an integral domain which is not a field.

Respectively, examples I chose were $\mathbb{Z}_5$ and $\mathbb{Z}[\sqrt{2}]$. Are these correct or do you have different/better examples?

Answer 1

1. $(\mathbb Z/5\mathbb Z)[X]$

2. $\mathbb Z$