Question about the zero ideal in $\mathbb{Z}$

Question

I have been studying Ring theory and a question came up.

Theorem: A commutative ring $R$ is integral domain iff the zero ideal $I=\{0\}$ is prime ideal in $R$.

Question: How can it be that in $\mathbb{Z}$ we do not count $I=\{0\}$ as a prime ideal ?(That's what is stated in my book)

We know that $\mathbb{Z}$ is integral domain. So the zero ideal must be prime.

Answer 1

An ideal is prime if whenever it contains a product it contains one of the factors. The zero ideal has that property just for integral domains.