I am trying to understand $\mathcal{O}_{\text{Spec A}}(D(f))$. This is the localization of the ring $A$ with respect to the multiplicative set of elements that are non-zero at all prime ideals that do not contain $(f)$. Here $f\in A$.
I can understand that the set $\{f,f^2,\dots\}$, in short all positive powers of $f$, are non-zero at the prime ideals not containing $(f)$. However, how do we know that this list is complete? Why can there not be other elements of $A$ that are non-zero at all prime ideals not containing (f)$?
Let $a$ be such an element; in other words $a$ is an element that is non-zero at all prime ideals not containing $(f)$, and $a\neq f^n$ for any natural number $n$. Does there have to be a prime ideal containing $a$ that does not contain $(f)$?
Assume that $I = (a)$ is disjoint from $\{1, f, f^2\ldots\}$. Apply Zorn lemma to get an ideal $J$ maximal to the condition "$I \subset J$ and $J$ is disjoint from $\{1, f, f^2\ldots\}$". Prove that $J$ is prime.