Left localizable rings and their characterizations

A new class of rings, the class of left localizable rings, is introduced. A ring R is left localizable if each nonzero element of R is invertible in some left localization S-1R of the ring R. Explicit criteria are given for a ring to be a left localizable ring provided the ring has only finitely many maximal left denominator sets (e.g., this is the case if a ring has a left Noetherian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets is a left localizable ring iff its left quotient ring is a direct product of finitely many division rings. A characterization is given of the class of rings that are finite direct product of left localization maximal rings.