The notion of the defining sheaf of ideals of a closed subscheme inspires the notion of defining ideal of a topologizing subcategory of an abelian category as the endofunctor which is the subfunctor of identity assigning to any the intersection of kernels of all morphisms where . One can show that if is an inclusion of topologizing subcategories, then .
If is an associative unital ring and a left ideal in . Let be the smallest coreflective subcategory of containing . Then is a two-sided ideal, namely the maximal 2-sided ideal in , which is explicitly, . Moreover, where is the square under the Gabriel multiplication agrees with .
See also conormal bundle.
A. L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9
V. A. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf