\begin{definition}[Definition:Closed Subscheme defined by Sheaf of Ideals]
Let $\struct {X, \OO_X}$ be a scheme.
Let $\II$ be a quasi-coherent sheaf of ideals of $\OO_X$.
Let $Y$ be the closed subset of $X$ defined by $\II$.
Let $\OO_Y := \OO_X / \II$ be the quotient sheaf of rings of $\OO_Y$ by $\II$.
Then the '''closed subscheme''' of $\struct {X, \OO_X}$ defined by $\II$ is:
:$\struct {Y, \OO_Y}$
\end{definition}