This is clearly a left ideal again. The special case is a two-sided ideal, namely the maximal ideal of contained in . If then we write .
A filter in the lattice of left ideals of a ring is a uniform filter if implies for any . Equivalently, the Gabriel composition of filters satisfies . The Gabriel composition of uniform filters is a uniform filter. Uniform filters are also called topologizing, because a non-empty set of left ideals of is a uniform filter iff it is the family of left ideals of which form an open neighborhood of in a “linear topology” on .
The uniform filters of ideals in a ring bijectively correspond to kernel functors on - (left exact subfunctors of the identity functor). The correspondence goes as follows. If is a uniform filter, and in -, define as the set of all such that is annihilated by some left ideal in . Conversely, given a kernel functor , define a uniform filter to be the filter whose members are all left ideals such that .
The most important class of uniform filters are Gabriel filters.