The ideal of polynomials vanishing over a point in an affine algebraic variety is maximal, and I think I understand the geometric intuition behind localizing it(s complement).
But what about multiplicative submonoids which are not complements of primes? In particular, what is the geometric intuition behind localizing at $S= \left\{ 1,a,a^2 ,\dots \right\}$?
Along the same line, what is the intuition behind the following fact?
Fact. $\sqrt{ \left\langle a \right\rangle}=\sqrt{ \left\langle b \right\rangle }\implies R[a^{-1}]\cong R[b^{-1}]$
Here $R[a^{-1}]$ denotes localization at $S$.