The largest strong left quotient ring of a ring

For an arbitrary ring R, the largest strong left quotient ring  Qls(R) of R and the strong left localization radical  lRs are introduced and their properties are studied in detail. In particular, it is proved that Qls(Qls(R))≃Qls(R), lR/lRss=0 and a criterion is given for the ring Qls(R) to be a semisimple ring. There is a canonical homomorphism from the classical left quotient ring Ql,cl(R)Ql,cl(R) to Qls(R) which is not an isomorphism, in general. The objects Qls(R) and lRs are explicitly described for several large classes of rings (semiprime left Goldie ring, left Artinian rings, rings with left Artinian left quotient ring, etc.).