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 $ℐ = ℐ_{S} \in End (A)$ 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 $ℐ_{S} \subset ℐ_{T}$ .

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 $ℐ_{T} (R) \subset R$ 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 $(J : R)^{2}$ .

nLab defining ideal of a topologizing subcategory

nLab
defining ideal of a topologizing subcategory