Localization of the Integer Ring Let $\mathbb{Z}$ be the ring of integers and let $p$ be a prime, then the $p$-localization of $\mathbb{Z}$ is defined as $\mathbb{Z}_{(p)}=\{\displaystyle\frac{a}{b}|a,b\in\mathbb{Z},p\nmid b\}$.
I can understand this definition literally but find it difficult to "see" what it really talks about. Would anyone help to shed more light on this definition, or recommend some further readings? 
 A: Intuitively, you can think about localization as a kind of 'zooming in' process on the prime $p$. The ring $\mathbb{Z}_{(p)}$ is like the ring $p$, but it is relatively more '$p$-focused'. This can be intuited in several ways. 
One possibility is that one can understand the information contained in a ring as being expressed in terms of obstructions. Every time an obstruction occurs, this is telling you some defect of the space it's defined on. When you localize $\mathbb{Z}$ at $(p)$, you get rid of a lot of the information (obstructions) coming from other points (primes) than $p$.
For example, the fact that $2$ is not invertible is an obstruction in $\mathbb{Z}$ which is, in some sense, coming from the point $(2)$. But, when you localize $\mathbb{Z}$ at a different point, say $(3)$, this obstruction disappears--$2$ is now invertible. 
So, what's left when you localize at $p$ is a ring which has forgotten information at all other points except for $(p)$ itself, from which you can conclude that you have 'zoomed in', or that the ring $\mathbb{Z}_{(p)}$ is a ring of functions on a space 'more local' to $(p)$. 
Of course, it's not 'super local' (they are only 'Zariski local'). There are still obstructions in $\mathbb{Z}$ coming from other primes, which are not 'fixed' (forgotten) by moving to $\mathbb{Z}_{(p)}$. For example, the existence of certain square roots, and more generally, higher order equations. 
To get rid of these obstructions, and in some sense, 'zoom in further' on the prime $(p)$, one must do a process called completion of $\mathbb{Z}_{(p)}$ at $(p)$. This zooms in further, and essentially only remembers the 'differential data' of $\mathbb{Z}_{(p)}$ at $(p)$--the only obstructions are coming from differential obstructions. But, I digress.
Hopefully that gives you a bit of insight as to how and 'see' $\mathbb{Z}_{(p)}$.
If you're interested in more of this line of thinking, I would suggest looking into the basics of algebraic geometry.
