This is the same question Writing a projective scheme as a union of irreducible subschemes. but this was not clearly answered: if I have $Z \subseteq \mathbb{P}^n_k$ a projective scheme, I can associate to that its ideal sheaf $I_Z$.
I've tried to prove that there exists irreducible closed subscheme $Z_i$ such that $I_Z=I_{Z_1}+\dots I_{Z_n}$, but I was not able to do that.
Locally on affine open sets we know how to do that. I tried to gluw what I got on open affines but because of the lack of uniqueness of primary decomposition I did not know how to make that.
If it can be any help, one can assume $Z$ to be equidimensional as far as I'm concerned.
