I am only just starting to come to terms with the definition of an affine scheme. I am still confused about what local sections of the sheaf are for open sets which aren't the distinguished basic open sets. Say we have a ring $A$ and set $X = \text{Spec}A$. For any basic open set, say $D(f)$, the sections of the sheaf $\mathcal{O}_{X}(D(f))$ look like elements of the localization $A_{f}$. From this, in principle, we should be able to use the gluing and identifying principle for sheaves to know what the sections look like for any open set.
Now suppose $U$ is not basic, but it is covered (for simplicity say be two) basic sets, $U = D(f) \cup D(g)$. My definition for a section of $\mathcal{O}_{X}(U)$ is a function, $$s: U \longrightarrow \bigsqcup_{\mathfrak{p} \in U}A_{\mathfrak{p}}, $$ where for every $p \in U$, there is a neighbourhood $\mathfrak{p} \in V \subseteq U$ such that $s$ looks like the ratio of elements of $A$ for all $\mathfrak{q} \in V$ where the denominator doesn't vanish at $\mathfrak{q}$ (i.e the denominator is not an element of the prime ideal $\mathfrak{q}$).
I think my question is a relatively simple one, I'm just missing some very fundamental concept, which I suspect is a direct corollary of the gluing and identifying principles. In particular, if we took the open neighbourhood $V$ above to be either $D(f)$ or $D(g)$ for each point, then it's obvious what the ratio is going to look like (it will be localized at powers of $f$ or $g$ respectively). But if I chose a different neighbourhood, say something properly contained in $D(f)$ or $D(g)$, or even something in their intersection, why does this give me the same sheaf? Is there ever a situation where you can't just say that the sections look like elements localized at powers of a single element? After all, the $D(f)$ form a cover.
Thanks
What you wrote is the general definition of $\mathcal O_X$ for $X=\mathrm{Spec} A$ (cf. Hartshorne, beginning of section II.2.) so you're right!
A sheaf $\mathcal F$ is uniquely determined by $\{\mathcal F(U) : U\in \mathcal B\}$, where $\mathcal B$ is a basis of the topology of $X$, as the family of distinguished open subsets. Since your definition gives the good result on these subsets, it gives the same sheaf as $\mathcal O_X$.