Normal Noetherian rings of dimension at least 1

Question

1. We want to pick $I_g$ to be "maximal" but what is the partial ordering to which it is maximal? For two $f,g \in A' - A$, I don't see how $I_f$ and $I_g$ would be related via subsets.

2. How can we see that $q \subseteq I_{a_2 g}$? $q = I_g = \{a \in A \mid ag \in A\}$ so $a_1 a_2 \in I_g \Rightarrow a_1 a_2 g \in A$ but $a_2 g \notin A$. Let $a$ be such that $ag \in A$. How can we see that $aa_2 g \in A$?

Answer 1

1. $I_g$ is maximal with respect to the inclusion of ideals.

2. If $ag\in A$ then $a(a_2g)\in A$ (since $a_2\in A$), so $\mathfrak q\subseteq I_{a_2g}$.