How do you prove the ideal $I= (X^2, XY)$ has infinitely many distinct irredundant primary decompositions?

Question

I have come up with the following two different decompositions of the ideal $I= (X^2, XY)$:
$I = (X) \cap (X^2, Y)$ and $I = (X) \cap (X^2, XY, Y^2) = (X) \cap (X, Y)^2$.

Can we generalize this somehow so that there are are infinitely many different primary decompositions?

Answer 1

Hint. $I=(X) \cap (X^2, XY, Y^n) $, for all $n\in \Bbb N$.