ag.algebraic geometry - Regular functions on affine schemes

I think a good reference might be Eisenbud-Harris, Geometry of Schemes. They construct the structure sheaf $mathcal{O}$ by specifying it on principal open subsets (viz. the 'important' ones) and extending it uniquely to other open subsets.

On a given ring $R$, you have a basis of open sets of Spec $R$ consisting of the $text{D}(f)$'s.

($D(f) = Spec R - V(Rcdot f)$, where $Rcdot f$ stands for the ideal generated by $f$).

With each $D(f)$ we associate the localization $R_f$.

With a general open subset $U$ we associate the inverse limit of the $R_f$, for $D(f) subseteq U$.

More concretely, if $U = Spec R - V(I)$, then $D(f) subseteq U$ if and only if $V(I) subseteq V(Rcdot f)$ if and only if $f in sqrt{I}$. So $mathcal{O}(U)$ is the inverse limit of the rings $R_f$, for $f in sqrt{I}$.

http://en.wikipedia.org/wiki/Inverse_limit