  We study a two-dimensional single band Hubbard Hamiltonian with antisymmetric
spin-orbit coupling. We argue that this is the minimal model to understand the
electronic properties of locally non-centrosymmetric transition-metal (TM)
oxides such as Sr$_2$IrO$_4$. Based on exact diagonalizations of small clusters
and the random phase approximation, we investigate the correlation effects on
charge and magnetic order as a function of doping and of the TM-oxygen-TM bond
angle $\theta$. For small doping and $\theta$ $\lesssim$ $15^\circ$ we find
dominant commensurate in-plane antiferromagnetic fluctuations while
ferromagnetic fluctuations dominate for $\theta$ $\gtrsim$ $25^\circ$.
Moderately strong nearest-neighbor Hubbard interactions can also stabilize a
charge density wave order. Furthermore, we compare the dispersion of magnetic
excitations for the hole-doped case to resonant inelastic X-ray scattering data
and find good qualitative agreement.
