  We have analysed the nature of persistent currents in open coupled mesoscopic
rings. Our system is comprised of two ideal loops connected to an electron
reservoir. We have obtained analytical expressions for the persistent current
densities in two rings in the presence of a magnetic field. We show that the
known even-odd parity effects in isolated single loops have to be generalised
for the case of coupled rings. We also show that when the two rings have
unequal circumferences, it is possible to observe opposite currents
(diamagnetic or paramagnetic) in the two rings for a given Fermi level.
