I want to show, that the intersection of two squarefree monomial ideals is again a squarefree monomial ideal. The definition of a squarefree monomial ideal I have is that the minimal set generating this ideal contains only squarefree monomial polynomials.
What I don't see is if $f \in G(I \cap J)$ (where I and J are squarefree monomial ideals and G(I) is the minimal generating set of ideal I) implies that $f\in G(I)$ or $f\in G(J)$. Than the statement would be obvious. So what is the relation of $G(I\cap J)$ and $G(I)\cap G(J)$?