We have the definition:
Definition. Let $X$ be a scheme. Let $Z,Y⊂X$ be closed subschemes corresponding to quasi-coherent ideal sheaves $\mathcal{I},\mathcal{J}⊂\mathcal{O}_X$. The scheme theoretic intersection of $Z$ and $Y$ is the closed subscheme of $X$ cut out by $\mathcal{I}+\mathcal{J}$. The scheme theoretic union of Z and Y is the closed subscheme of $X$ cut out by $\mathcal{I}∩\mathcal{J}$.
Is true, in general, that given closed subschemes $Y,Z$ and $W$ of $X$ we have
$$Y\cap (Z\cup W)= (Y\cap Z)\cup (Y\cap W), \mbox{scheme-theoretically?}$$
If not, there is any necessary and sufficient conditions? Any hint, reference or solution is welcome!!
Remark: Using the language of ideal, if we denote $I_*$ the ideal of the variety $*$ we have just one inclusion, in general $$I_Y+(I_Z\cap I_W)\subseteq (I_Y+I_Z)\cap(I_Y+I_W).$$