I can understand what type of ring we obtain when localizing at a prime ideal of height 1, but I can't imagine what type of ring I obtain for greater heights.
A simple example I don't comprehend is localizing $\mathbb{Z}[x]$ at the maximal ideal $\mathfrak{m}=(2,1+x).$ What are the prime ideals of $\mathbb{Z}[x]$ that are included in $\mathfrak{m}$? Maybe a scheme-geometric illustration could help to understand the ring $\mathbb{Z}[x]_\mathfrak{m}.$
Thanks!
There are many,many prime ideals included in $\mathfrak m$. For example:
$$(0),(2), (x+1),(x-1),(x^2+1),(x^2+123456789),(x^3+3x^2+3x+3),\cdots$$ The last ideal (for example) is prime because it is generated by an irreducible polynomial (Eisenstein) and it is included in $\mathfrak m=(x+1,2)$ because $x^3+3x^2+3x+3=(x+1).(x+1)^2+2.1$