Of the several types of
noncommutative quotient rings, finite left localizations have structure most like that
of the original ring. This paper examines finite left localizations from two points of
view: As rings of quotients with respect to hereditary torsion classes, and as
endomorphism rings of finitely generated projective modules. In the first case, finite
left localizations are shown to be the rings of quotients with respect to perfect
TTF-classes. In the second, they are shown to be the double centralizers of finite
projectors.
