About Primary Decomposition - $\displaystyle\cup^n_{i=1}\mathfrak{p}_i=\{x\in A:(\mathfrak{a}:x)\neq\mathfrak{a}\}$

Question

I am studying, on Commutative Algebra, this proposition, from Atiyah and MacDonald (page 53)

I understood the proof, but I am unsure to transpose the proof from zero ideal to the general case $\displaystyle\cup^n_{i=1}\mathfrak{p}_i=\{x\in A:(\mathfrak{a}:x)\neq\mathfrak{a}\}$.

Thanks in advance for some light!

Answer 1

What you need to show is that everything involved in the theorem respects taking the inverse image under the map $\pi : A \to A/\mathfrak{a}$.

If you have trouble showing something respects this map, let me know and I'm happy to give more details!