Let be a (possibly noncommutative associative) ring (unital or not). The collection of one sided (say right) ideals has a partial order with respect to inclusion; a filter with respect to this partial order is a set of right ideals such that ; if , are right ideals, and then ; and if then .
A filter of ideals in is topologizing if
it is uniform, i.e. for any and , the right ideal
is in .
if and for all then .
The term topologizing is explained by the following statement:
A topologizing set of right ideals in a ring is a basis of neighborhoods of for a topology on .
Carl Faith, Algebra Vol. I, page 520
Harold Simmons, The semiring of topologizing filters of a ring, doi