In our definition this notion is equivalent to topologizing filter; though for some authors the latter notion slightly differs. Stenstroem says Gabriel topology instead of Gabriel filter, because all Gabriel filters form a basis of nieghborhoods of for a topology on .
If and are left ideals in a Gabriel filter , then the set (of all products where ) is an element on . Any uniform filter is contained in a minimal Gabriel filter (said to be generated by ), namely the intersection of all Gabriel filters containing . Given a Gabriel filter , the class of all -torsion modules (see uniform filter) is a hereditary torsion class.