  We carry out a comprehensive analysis of the behavior of the
magnetorotational instability (MRI) in viscous, resistive plasmas. We find
exact, non-linear solutions of the non-ideal magnetohydrodynamic (MHD)
equations describing the local dynamics of an incompressible, differentially
rotating background threaded by a vertical magnetic field when disturbances
with wavenumbers perpendicular to the shear are considered. We provide a
geometrical description of these viscous, resistive MRI modes and show how
their physical structure is modified as a function of the Reynolds and magnetic
Reynolds numbers. We demonstrate that when finite dissipative effects are
considered, velocity and magnetic field disturbances are no longer orthogonal
(as it is the case in the ideal MHD limit) unless the magnetic Prandtl number
is unity. We generalize previous results found in the ideal limit and show that
a series of key properties of the mean Reynolds and Maxwell stresses also hold
for the viscous, resistive MRI. In particular, we show that the Reynolds stress
is always positive and the Maxwell stress is always negative. Therefore, even
in the presence of viscosity and resistivity, the total mean angular momentum
transport is always directed outwards. We also find that, for any combination
of the Reynolds and magnetic Reynolds numbers, magnetic disturbances dominate
both the energetics and the transport of angular momentum and that the total
mean energy density is an upper bound for the total mean stress responsible for
angular momentum transport. The ratios between the Maxwell and Reynolds
stresses and between magnetic and kinetic energy densities increase with
decreasing Reynolds numbers for any magnetic Reynolds number; the lowest limit
of both ratios is reached in the ideal MHD regime.
