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!