Let p be prime ideal in the ring R. I want to show that there is a bijection between the set of prime ideals of R and the set of prime ideals of localization of R.
I am quite confused about the localization of a ring. How do the elements of localization of R look like and how do its prime ideals look like?
Anything that's not in $p$ becomes invertible in $R_p$, so any ideal $a$ with $a \not\subseteq p$ disappears (it becomes the whole ring, since it contains units). Most importantly, any ideal $a$ with $a \supsetneq p$ disappears, which means that $p$ is a maximal ideal, and in fact the only maximal ideal. The localization therefore creates a local ring (funny how that works). Any ideal contained in $p$ is still there, intact, though.
In some sense, when looking at ideals, it's the opposite of the quotient $R/p$, where all ideals that are left are the ones that contains $p$, while in the localization, all ideals that are left are the ones contained in $p$.
