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?

What is the ring of this ideal? Does being able to be decomposed in infinitely many ways have any relation with the ring on which the ideal is defined? — Tedebbur, Jul 07 '21 at 00:59

score 2 · Accepted Answer · answered Mar 08 '16 at 07:59

2

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

answered Mar 08 '16 at 07:59

user 1

7,652

This answer is helpful. Thanks. – Mark Mar 08 '16 at 08:02
you are welcome – user 1 Mar 08 '16 at 16:16
@user 1 Can this(or any other) primary decomposition be uncountably infinite? – Tedebbur Jul 07 '21 at 00:53

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

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