  We study the two-dimensional Hubbard model with the Rashba type spin-orbit
coupling within and beyond the mean-field theory. The antiferromagnetic ground
state for the model at half-filling and the Cooper pairing induced by
antiferromagnetic spin fluctuations near half-filling are examined based on the
random-phase approximation. We show that the antiferromagnetic order is
suppressed and the magnetic susceptibility turns out to be anisotropic in the
presence of the spin-orbit coupling. Energy spectrums of transverse spin
fluctuations are obtained and the effective interactions between holes mediated
by antiferromagnetic spin fluctuations are deduced in the case of low hole
doping. It seems that the spin-orbit coupling tends to form s+p-wave Cooper
pairs, while the s+d-wave pairing is dominant when the spin-orbit coupling is
absent.
