  For an arbitrary ring $R$, the largest strong left quotient ring $Q_l^s(R)$
of $R$ and the strong left localization radical $\glsR$ are introduced and
their properties are studied in detail. In particular, it is proved that
$Q_l^s(Q_l^s(R))\simeq Q_l^s(R)$, $\gll^s_{R/\glsR}=0$ and a criterion is given
for the ring $ Q_l^s(R)$ to be a semisimple ring. There is a canonical
homomorphism from the classical left quotient ring $Q_{l, cl}(R)$ to $Q_l^s(R)$
which is not an isomorphism, in general. The objects $Q_l^s(R)$ and $\gll^s_R$
are explicitly described for several large classes of rings (semiprime left
Goldie ring, left Artinian rings, rings with left Artinian left quotient ring,
etc).
