Some Remarks on Categories of Modules modulo Morphisms with Essential Kernel or Superfluous Image

For an ideal I of a preadditive category A, we study when the canonical functor C : A → A/I is local. We prove that there exists a largest full subcategory C of A for which the canonical functor C : C → C/I is local. Under this condition, the functor C turns out to be a weak equivalence between C and C/I. If A is additive (with splitting idempotents), then C is additive (with splitting idempotents). The category C is ample in several cases, such as the case when A = Mod-R and I is the ideal ∆ of all morphisms with essential kernel. In this case, the category C contains, for instance, the full subcategory F of Mod-R whose objects are all the continuous modules. The advantage in passing from the category F to the category F/I lies in the fact that, although the two categories F and F/I are weakly equivalent, every endomorphism has a kernel and a cokernel in F/∆, which is not true in F . In the final section, we extend our theory from the case of one ideal I to the case of n ideals I1, . . . , In.