For an ideal

of a preadditive category

, we study when the canonical functor

is local. We prove that there exists a largest full subcategory

of

, for which the canonical functor

is local. Under this condition, the functor

, turns out to be a weak equivalence between

, and

. If

is additive (with splitting idempotents), then

is additive (with splitting idempotents). The category

is ample in several cases, such as the case when

=Mod-R and

is the ideal

of all morphisms with essential kernel. In this case, the category

contains, for instance, the full subcategory

of Mod-R whose objects are all the continuous modules. The advantage in passing from the category

to the category

lies in the fact that, although the two categories

and

are weakly equivalent, every endomorphism has a kernel and a cokernel in

, which is not true in

. In the final section, we extend our theory from the case of one ideal

to the case of

ideals

,

,

.