On finite left localizations

Abstract

Of the several types of noncommutativ e 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. The first characterization (Corollary 1.2) shows that finite left localizations may be realized as endomorphism rings of finitely generated projective idempotent ideals. It follows that every left localization of a right perfect ring is finite. Finally, the finite projectors of the second characterization (Corollary 2.2) are shown to have endomorphism rings whose quotient ring structure is very closely related to that of the original ring. For example, every ring of left quotients of the endomorphism ring is Morita equivalent to a ring of quotients of the original ring.