# nLab defining ideal of a topologizing subcategory

The notion of the defining sheaf of ideals of a closed subscheme inspires the notion of defining ideal of a topologizing subcategory $S$ of an abelian category $A$ as the endofunctor $\mathcal{I}=\mathcal{I}_S\in End(A)$ which is the subfunctor of identity $Id_A$ assigning to any $M\in A$ the intersection of kernels $Ker(f)$ of all morphisms $f: M\to N$ where $N\in Ob(S)$. One can show that if $T\subset S$ is an inclusion of topologizing subcategories, then $\mathcal{I}_{S}\subset \mathcal{I}_{T}$.

If $R$ is an associative unital ring and $J\subset R$ a left ideal in $R$. Let $T = [J]$ be the smallest coreflective subcategory of $A$ containing $R/J$. Then $\mathcal{I}_T(R)\subset R$ is a two-sided ideal, namely the maximal 2-sided ideal in $J$, which is explicitly, $(J:R) = \{r\in R\,|\, J r \subset J\}$. Moreover, $J_{T\circ T}$ where $T\circ T$ is the square under the Gabriel multiplication agrees with $(J:R)^2$.