What is the ideal sheaf of a closed subset of a scheme?

Question

Let $(X, \mathcal{O}_{X})$ be a scheme and let $Y \subseteq X$ be a closed subset. What is meant by the "ideal sheaf of the closed subset $Y$"? Normally to define the ideal sheaf of a closed subscheme we take the kernel of the morphism $$ i^{\#}: \mathcal{O}_{X} \longrightarrow i_{*}\mathcal{O}_{Y}. $$ But in Hartshorne III Theorem 3.7, we simply have a scheme $X$ with a closed subset $Y$, and he defines the ideal sheaf of $Y$. But $Y$ has no scheme structure, so if we want to define it as above, what is meant by $\mathcal{O}_{Y}$? Indeed we could put many different scheme structures on $Y$. Is there some more general definition of the ideal sheaf if we only specify a closed subset?