I'm reading Bhatt and Scholze's, "Prisms and prismatic cohomology", 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", 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 $Spec(A)$. 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 $Spec(A)$.
   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.

I'm reading Bhatt and Scholze's, "Prisms and prismatic cohomology", 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", 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 $Spec(A)$. On the other hand, seeing Def. 2.1.12, ibid., regarding Zariski localizations of spectral spaces, I imagine it should be something like $$\sqcup_{\mathfrak p \in V(I)} Spec(A_{\mathfrak p})$$ endowed with the natural map to $Spec(A)$.
   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.

I'm reading Bhatt and Scholze's, "Prisms and prismatic cohomology", 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", 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 $Spec(A)$.
   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.

I'm reading Bhatt and Scholze's, "Prisms and prismatic cohomology", 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", 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 $Spec(A)$. 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 $Spec(A)$.
   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.