  Electron transport through multi-terminal rectangular arrays of quantum rings
is studied in the presence of Rashba-type spin-orbit interaction (SOI) and of a
perpendicular magnetic field. Using the analytic expressions for the
transmission and reflection coefficients for single rings we obtain the
conductance through such arrays as a function of the SOI strength, the magnetic
flux, and of the wave vector $k$ of the incident electron. Due to destructive
or constructive spin interferences caused by the SOI, the array can be totally
opaque for certain ranges of $k$, while there are parameter values where it is
completely transparent. Spin resolved transmission probabilities show
nontrivial spin transformations at the outputs of the arrays. When point-like
random scattering centers are placed between the rings, the Aharonov-Bohm peaks
split, and an oscillatory behavior of the conductance emerges as a function of
the SOI strength.
