# quotient ideal & primary decomposition

A 'quotient ideal' associated to a pair of ideals $$\frak{a}, \frak{b}$$ $$\subset R$$ of a commutative ring with $$1_R$$ is a new ideal defined as $$(\frak{a}:\frak{b})$$ $$= \{r \in R \mid r\frak{b} \subset \frak{a} \}$$. at wikipedia page about quotient ideals I found a remark that needs clarification. the assertion is that the ideal quotient is useful for calculating primary decompositions.

How concretly the ideal quotient helps to determine a primary decomposition of a ideal? let me remind that a primary decomposition of a ideal $$\frak{a}$$ is if we can write this ideal as an intersection $$\frak{a}= P_1 \cap P_2 \cap ... \cap P_m$$ where $$\frak{P}_i$$ are primary ideals. I would be very grateful if somebody could explain the main idea why the quotient ideals provide a useful tool to calculate such primary decomposition.

One such example is Atiyah-Macdonald Theorem 4.5. If we assume that the decomposition is minimal, then the prime ideals $$\mathfrak{p}_i = r(\mathfrak{P}_i)$$ are precisely the ideals $$r(\mathfrak{a} : x)$$ for $$x \in R$$.