I am trying to collect all the relations between union and intersection This far I have proved
- $\cap I(W_i) = I(\cup W_i)$
- $\cap V(I_i)=V(\cup I_i)=V(\sum I_i)$ where $<\cup I_i>=\sum I_i$
- $\cup V(I_i)=V(\prod I_i)$ where $\prod I_i=\cap I_i$ if the ideals are comaximal (i think here $I$ has to be finite)
- What about $\cup I(W_i) = I(\cap W_i)$? It is clear that $\cup I(W_i) \subseteq I(\cap W_i)$ so, $\sum I(W_i) \subseteq I(\cap W_i)$. But can we say something more?
Next, Let $X=V(a)\subseteq A^n$ be an algebraic set. TFAE:
- $X$ is an irreducible topological space;
- the ideal $\mathrm{rad}(a)$ is prime.