  We discuss how one can realize a photonic device that combines synthetic
dimensions and synthetic magnetic fields with spatially local interactions.
Using an array of ring cavities, the angular coordinate around each cavity
spans the synthetic dimension. The synthetic magnetic field arises as the
intercavity photon hopping is associated with a change of angular momentum.
Photon-photon interactions are local in the periodic angular coordinate around
each cavity. Experimentally observable consequences of the synthetic magnetic
field and of the local interactions are pointed out.
