  We analyze a detailed model of a Bose-Einstein condensate trapped in a ring
optical resonator and contrast its classical and quantum properties to those of
a Fabry-P{\'e}rot geometry. The inclusion of two counter-propagating light
fields and three matter field modes leads to important differences between the
two situations. Specifically, we identify an experimentally realizable region
where the system's behavior differs strongly from that of a BEC in a
Fabry-P\'{e}rot cavity, and also where quantum corrections become significant.
The classical dynamics are rich, and near bifurcation points in the mean-field
classical system, the quantum fluctuations have a major impact on the system's
dynamics.
