I'm reading Bhatt and Scholze's, ["Prisms and prismatic cohomology"](https://arxiv.org/pdf/1905.08229.pdf), and I have a few questions about Lemma 3.1.

1. Why can they choose a finite number of elements generating $A$ such that $IA[1/g_i]$ is principal? I don't understand how this follows from the fact that $I$ is locally principal.
2. What do they mean by the Zariski localization of a ring along a closed subset of its spectrum?
From Def. 2.1.6 in Bhatt and Scholze's, ["The pro-étale cohomology for schemes"](http://www.math.uni-bonn.de/people/scholze/proetale.pdf), I understand that the __localization of $Spec(A)$ along $V(I)$__ should be something like
$$\cup_{\mathfrak p \in V(I)} Spec(A_{\mathfrak p}),$$ 
which can be seen as a subset of $Spec(A)$ via the open immersions $Spec (A_{\mathfrak p}) \hookrightarrow Spec(A)$. On the other hand, seeing Def. 2.1.12, ibid., regarding __Zariski localizations of spectral spaces__, I immagine it should be something like 
$$\sqcup_{\mathfrak p \in V(I)} Spec(A_{\mathfrak p})$$ 
endowed with the natural map to $Spec(A)$. <br/>
And so the Zariski localization of $A$ along $V(I)$ should be something like 
$$\prod_{\mathfrak p \in V(I)} A_{\mathfrak p},$$ 
but I don't know how this fits Def. 2.2.1, ibid., regarding __Zariski localizations of rings__.