Given a field $k$, is it always the case that $I(A\cup B)=I(A)\cap I(B)$ for any two $A,B\subseteq k^n$ where $I(A)=\{p\in k[x_1,\dots,x_n]:\forall x\in A\;\;p(x)=0\}$?
the "$\subseteq$" is obvious because $A\mapsto I(A)$ is decreasing and $A,B\subseteq A\cup B$. If $p\in I(A)\cap I(B)$ then $\forall x\in A\cup B$ we have $x\in A$ or $x\in B$ so $p(x)=0$ and then $p\in I(A\cup B)$, so $I(A)\cap I(B)\subseteq I(A\cup B)$
The proof seems correct to me but i cant find this equation anywhere so it may be wrong. If so, where?
Thanks