An infinite number of different decompositions $I = \cap_{i} q_i$ of ideal in polynomial ring $k[x, y, z]$ where $q_i$ are primary ideals

Question

Suppose we have an ideal $I \subset k[x, y, z]$ of a polynomial ring of three variables. Can i have an infinite number of different decompositions $I = \cap_{i} q_i$ where $q_i$ are primary ideals, which are associated with $p_i \in \text{Ass}_I(A/I)$? I couldn't find an example, so may be it's impossible, but i couldn't prove it.

Answer 1

The standard example is $I=(x^2,xy)\subseteq k[x,y,z]$. Given $n\geq 1$, we can write $$I=(x)\cap (x^2,xy,y^n).$$ See Atiyah-MacDonald, Exercise 8.1 for more theory.