Why can't closed subschemes be defined in an easier way?

Question

My (naive) definition of closed subscheme of a scheme $X$ would be: a closed subset $Z \overset{j}{\hookrightarrow} X$ endowed with the structure sheaf $\mathcal{O}_Z := j^{-1}\mathcal{O}_X$. Apparently, though, this is too narrow a definition, and we need to allow for structure sheaves of the form $j^{-1} \left( \mathcal{O}_X / \mathcal{I} \right)$, where $\mathcal{I} \subseteq \mathcal{O}_X$ is a sheaf of ideals. Can anybody explain to me why this is needed?