  We consider a two-dimensional electron system subjected to a short-ranged
nonmagnetic disorder potential, Coulomb interactions, and Rashba spin-orbit
coupling. The path-integral approach incorporated within the Keldysh formalism
is used to derive the kinetic equation for the semiclassical Green's function
and applied to compute the spin current within the linear response theory. We
discuss the frequency dependence of the spin Hall conductivity and further
elucidate the role of electron interactions at finite temperatures for both the
ballistic and diffusive regimes of transport. We argue that interaction
corrections to the spin Hall effect stem from the quantum interference
processes whose magnitude is estimated in terms of parameters of the considered
model.
