Let $(X,O_X)$ be a scheme, and $Y$ be a Zariski closed subset. For any $U$ we define $I(U)$ to be the subset of $O_X(U)$ which vanish on $U\cap Y$.
My main question is what does this even mean? I initially thought that $I(U)$ was the kernel of the restriction map $\operatorname{res}^U_{U\cap Y}$ but then realized that doesn't make sense since $U\cap Y$ is not necessarily open in $X$, so what does it mean for an element to vanish on $U\cap Y$? If I was dealing with a sheaf of say continuous functions to $\mathbb{R}$, then there is a nice evaluation map that makes sense, but since the ring $O_X(U)$ can't be identified in such a way, I am at a a loss of what to do. Once I know that I am sure I can show that $I(U)$ is an ideal, I just don't get what this means.