Non-uniqueness of embedded primary components

Question

I'm studying the Uniqueness theorems for primary decomposition of decomposable ideals in a ring. In my notes, after having proved the two classical theorems, there is the following remark:

Remark. In the light of the preceding results, the isolated primary components (i.e. the primary components $\mathfrak{q}_i$ corresponding to minimal prime ideals $\mathfrak{p}_i$) are uniquely determined by the ideal $\mathfrak{a}$ itself. The embedded primary components are not, in fact there are infinite ways to choose each embedded component.

So I'm trying to understand the last statement, i.e. "There are infinite ways to choose each embedded component". It seems to me to be a quite strong non-uniqueness specular result. Why is it so? More specifically, given a decomposable ideal $\mathfrak{a}$, a minimal primary decomposition and an embedded component of the latter:

• Is there a "canonical" way to define a single different (and suitable) embedded component from the one given?
• Can we prove the existence of an infinite amount of alternative possible choices for the given embedded component? Under which conditions?

Thanks

Answer 1

I can answer you to the second question: If A is Noetherian. The details are given in Atiyah-Macdonald's "Introduction to Commutative Algebra", Problem 1 of Chapter 8.