Write out a minimal primary decomposition of the ideal $I = (XY, Y Z, XZ) ⊆ \mathbb C[X, Y, Z]$, and determine the primes belonging to $I$. Determine the dimension of the ring $\mathbb C[X, Y, Z]/(XY, Y Z, XZ)$.
If the decomposition of ideal $I$ is given then the proof is straightforward. However, without any pre-information, how would I find a primary decomposition of the ideal $I$? Can anyone give me some insight? Thank you in advance !
