Can a Ringworld have a 'moon'?

The idea is to have a number of artificial, spherical satellite to have sort of a "spiral" orbit around a Ringworld, such that the inhabitants of each of a number of ring sections see a moon in the sky once every year or two. Is that theoretically possible for a sufficiently advanced civilization?

Edit: By 'Ringworld' I mean a Niven-Ring, I should've clarified that sooner and I will try to do better next time. Thank you all for your answers! I think I have a better idea of what I want to do now.

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Can a Ringworld have a 'moon'?

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