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.
Best Answer
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 $x \in R$.
Other applications appear in Chapter 4 of Atiyah-Macdonald.