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I have a question regarding, how to find/calculate the minimal primary decomposition given an ideal $\mathfrak{a}$. In my case, let $k$ be a field. Consider the polynomial ring $A:= k[X,Y,Z]$ and the ideal $\mathfrak{a} := (X^2, XY, YZ, XZ) \subset A$. According to the solution is $$\mathfrak{b}:= (X,Y) \cap (X,Z) \cap (X^2,Y^2, Z^2, XY, XZ, YZ)$$ a minimal primary decomposition of $\mathfrak{a}$, but I don't see why. We have always worked with the properties of such MPD, but never had to compute it explicitly, so I don't know how one comes to this solution.

I have tried to understand it using this link on stackexchange . But I got stuck, when I tried to implement it for this example. I would be thankful, if someone could explain me, how one does this.

Thanks in advance!

user26857
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MBCLA
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    Your ideal is monomial, and in this case there is an algorithm for finding a primary decomposition; see this answer. – user26857 Jan 03 '18 at 18:48
  • I have tried to solve it the same way, but I don't come to the last decomposition with all those powers. Can you see where the mistake is? $(X^2, XY, YZ, XZ) = (X^2, XY, YZ, X) \cap (X^2, XY, YZ, Z) = (X^2, XY, Y, X) \cap (X^2, XY, Z, X) \cap (X^2, XY, Y, Z) \cap (X^2, XY, Z) = (X,Y) \cap (X,Y) \cap (X,Z) \cap (X^2, Y,Z,X) \cap (X^2, X,Y,Z) \cap (X^2,Y,Z) \cap (X,Z) \cap (X^2,Y,Z)$.

    This is what I have found out so far, but I don't come to the last decomposition as above. Do you have any hints for me @user26857 ?

     – MBCLA Jan 03 '18 at 19:11
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    You can simplify the things on the way. For instance, $(X^2, XY, YZ, X)=(X,YZ)=(X,Y)\cap(X,Z)$. – user26857 Jan 03 '18 at 19:16
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    One more thing: the last member of the decomposition isn't unique since that primary ideal is embedded. So you don't need to get at the same decomposition. As you can see we immediately get: $(X^2, XY, YZ, XZ)=(X,Y)\cap(X,Z)\cap(X^2,Y,Z)$ – user26857 Jan 03 '18 at 19:19
  • So the MPD is not always (or never) unique? Thus, if I continue my calculation, I get: $(X,Y) \cap (X,Y) \cap (X,Z) \cap (X^2,Y,Z) \cap (X^2,Y,Z) \cap (X^2,Y,Z) \cap (X,Z) \cap (X^2,Y,Z) = (X,Y) \cap (X,Z) \cap (X^2,Y,Z)$ as you have written.

    But for understanding, can you guess how they come to this particular last decomposition in $\mathfrak{b}$?

     – MBCLA Jan 03 '18 at 19:36
  • The embedded primary ideals are not necessarily unique, so there could be different minimal primary decompositions. In this case they wanted the last primary ideal to be $(X,Y,Z)^2$, probably in order to see very clear that it is primary (as having the radical a maximal ideal). I can guess how they find it, but I'm not sure that my guess is the way they thought. – user26857 Jan 03 '18 at 19:53
  • It would be great if you could tell me your guess. If I'm at any time later being asked to write down two MPDs of the same ideal, so I might have a clue, where to start. – MBCLA Jan 03 '18 at 20:37

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