$\mathbb Z$ as an example of scheme

Question

The idea behind Grothendieck's concept of schemes is that given a ring $R$ we can see $R$ as the ring of regular function over a topological space $\operatorname{Spec}(R)$. But I have difficulties to grasp this idea due to its abstraction. For example take $R=\mathbb Z$ the ring of integers - that will be the ring of of regular function over a topological space which is $\operatorname{Spec}(\mathbb Z)=\{p\mathbb Z : p \text{ is a prime number}\}$ so how is it possible to see $\mathbb Z$ as ring of function?

Answer 1

You can think of an element $n\in \mathbb{Z}$ as a 'function' $\phi_n$ on $\operatorname{Spec} \mathbb{Z} $ as follows:

For every $p\mathbb{Z} \in \operatorname{Spec}\mathbb{Z} $, consider $\phi_n(p\mathbb{Z} )=\bar{n}\in \mathbb{Z} /p\mathbb{Z} =\kappa(p)$.

Here $\kappa(p)$ denoted the 'residue field' of the point $p$.