We report on a magnetophotoluminescence study of single self-assembled semiconductor nanorings which are fabricated by molecular-beam epitaxy combined with AsBr3 in situ etching. Oscillations in the neutral exciton radiative recombination energy and in the emission intensity are observed under an applied magnetic field. Further, we control the period of the oscillations with a gate potential that modifies the exciton confinement. We infer from the experimental results, combined with calculations, that the exciton Aharonov-Bohm effect may account for the observed effects.