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I am looking for a minimal primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$.

I realize that this is a similar question to some of the previous ones, but the ring is different than in those questions and I would like to both be sure that I get this right in the end and, more importantly, ask if there is a method to this madness. Namely, in the case of monomial ideals there is a very simple algorithm that has been previously posted here. I realize that it can't be that simple in this case but if there are some helpful tricks that can be used that would be good to know.

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baltazar
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    Hint: This particular example has the helpful property that $Z = XY$, so the ring is isomorphic to $k[X,Y] / (X Y^2, X^3 Y)$. – Jake Levinson Jan 26 '15 at 00:16
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    @JakeLevinson Having found the decomposition of $(0)$ in $k[X,Y]/(XY^2,X^3Y)$ to be $(Y) \cap (X) \cap (X^3,Y^2)$, how might one get back the equivalent thing in the original ring? – baltazar Jan 26 '15 at 00:32
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    In this case you can't say the minimal primary decomposition since the ideal has an embedded prime. – user26857 Jan 26 '15 at 00:59
  • @user26857..sorry how do we know that it has an embedded prime?(without computing primary decomposition) – Arpit Kansal Oct 19 '15 at 10:12

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In fact, you are looking for a minimal primary decomposition of $I=(X^2Z,YZ,Z-XY)$ in $k[X,Y,Z]$.

As it is noticed in the comments, $k[X,Y,Z]/(Z-XY)\simeq k[X,Y]$. The image of $I/(Z-XY)$ under this isomorphism is $(X^3Y,XY^2)$ and a primary decomposition of this ideal is given by $(X)\cap(Y)\cap (X^3,Y^2)$. The inverse image of this decomposition is $$(X,Z-XY)/(Z-XY)\cap(Y,Z-XY)/(Z-XY)\cap(X^3,Y^2,Z-XY)/(Z-XY)=$$ $$(X,Z)/(Z-XY)\cap(Y,Z)/(Z-XY)\cap(X^3,Y^2,Z-XY)/(Z-XY),$$ and thus $I=(X,Z)\cap(Y,Z)\cap(X^3,Y^2,Z-XY)$.

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