  A new class of rings, {\em the class of weakly left localizable rings}, is
introduced. A ring $R$ is called {\em weakly left localizable} if each
non-nilpotent element of $R$ is invertible in some left localization $S^{-1}R$
of the ring $R$. Explicit criteria are given for a ring to be a weakly left
localizable ring provided the ring has only finitely many maximal left
denominator sets (eg, this is the case if a ring has a left Artinian left
quotient ring). It is proved that a ring with finitely many maximal left
denominator sets that satisfies some natural conditions is a weakly left
localizable ring iff its left quotient ring is a direct product of finitely
many local rings such that their radicals are nil ideals.
