We investigate the exciton energy level structure of a large ensemble of InAs/GaAs quantum rings by photoluminescence spectroscopy in magnetic fields up to 30 T for different excitation densities. The confinement of an electron and a hole in these type I quantum rings along with the Coulomb interaction suppress the excitonic Aharonov-Bohm effect. We show that the exciton energy levels are nonequidistant and split up in only two levels in magnetic field, reflecting the ringlike geometry. A model, based on realistic parameters of the self-assembled quantum rings, allows us to interpret the essential features of the observed PL spectra in terms of the calculated optical transition probabilities.