I'm reading through Siegfried Bosch's Commutative Algebra book, and I'm confused on his notation in one his proofs. He uses this notation a lot, so I think I should I understand it. The notation first appears in his proof of Theorem 8:
Theorem 8. For a decomposable ideal $\mathfrak{a} \subset R$, let $\mathfrak{a} = \bigcap_{i=1}^r \mathfrak{q_i}$ be a minimal primary decomposition. Then the set of corresponding prime ideals $\mathfrak{p}_i = \text{rad}(\mathfrak{q}_i)$ coincides with the set of all prime ideals in $R$ that are of type $\text{rad}\big((\mathfrak{q}:x)\big)$ for $x$ varying over $R$.
In his proof, he writes: Let $x\in R$. The primary decomposition of $\mathfrak{a}$ yields $(\mathfrak{a}:x) = \bigcap_{i=1}^r(\mathfrak{q_i}:x)$ and therefore by Lemma 7, a primary decomposition
$ \text{rad}\big((\mathfrak{a}:x)\big) = \bigcap_{i=1}^r\text{rad}\big((\mathfrak{q}_i:x)\big) = \bigcap_{x\not\in \mathfrak{q}_i}\mathfrak{p}_i. $
What I'm confused about is that last equality, what's that mean? The indexing confuses me. If it's helpful, here's the part of Lemma 7 I think he's using.
Lemma 7. Consider a primary ideal $\mathfrak{q}\subset R$ and the corresponding prime ideal $\mathfrak{p} = \text{rad}(\mathfrak{q})$. Then, for $x\not\in \mathfrak{q}$, $(\mathfrak{q}:x)$ is $\mathfrak{p}$-primary.
Thank you.