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The primary decomposition of an ideal $(x^2, xy)$ is $$(x^2, xy) = (x) \cap (x, y)^2$$ which can be found on these notes.

Could someone explain to me how this can be done?

Edited: My question is not restricted to the example I gave $(x^2, xy)$, one can use any other example to show me the process.

user26857
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marya
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  • Related: http://math.stackexchange.com/questions/990898/primary-decomposition-of-x2-xy-xz-in-kx-y-z-where-k-is-a-field, http://math.stackexchange.com/questions/1688151/how-do-you-prove-the-ideal-i-x2-xy-has-infinitely-many-distinct-irredunda – user26857 May 21 '16 at 10:17

1 Answers1

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Answer 1.In general, to see an illustration of primary decomposition:
The book "Monomial Ideals" by "Herzog-Hibi", section 1.3 (Primary decomposition and associated prime ideals) illustrate this. The book has:
"Theorem 1.3.1 in combination with Corollary 1.3.2 now says that each monomial ideal has a unique presentation as an irredundant intersection of irreducible (primary) monomial ideals. The proof of Theorem 1.3.1 shows us how we can find such a presentation." Example 1.3.3 illustrates the procedure.
I add an image of Example 1.3.3 and highlight the first step:
enter image description here

So we have $(x^2, xy) = (x^2, x)\cap (x^2,y) = (x) \cap (x^2, y)$

Answer 2. For your special example:
Let K be a field and let R = K[X,Y]. Let M=(X,Y). The book "Steps in Commutative Algebra" by "Sharp", shows (Example 4.27, page 74) that $I=(x^2, xy) = (x) \cap (x, y)^2=(x) \cap (x^2, y)$. So the primary decomposition of "answer 1" and the primary decomposition you want are both primary decompositions for $I$.

user 1
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    In this case the answer would be $$(x^2, xy) = (x^2,x) \cap (x^2, y)=(x)\cap(x^2,y)$$ (It is fine but) why do we need to replace $(x^2,y)$ by $(x,y)^2=(x^2,xy,y^2)$ – marya May 21 '16 at 08:19
  • 1- primary decomposition is not unique. 2- we dont replace $(x^2,y)$ by $(x^2,xy,y^2)$, but we replace $(x)\cap (x^2,y)$ by $ (x)\cap (x^2,xy,y^2)$. 3- we dont need to replace. + give a link for "which can be found all over this site" – user 1 May 21 '16 at 08:45
  • There are even more decompositions – user 1 May 22 '16 at 07:58