That's a tricky one. Do you mean make the ring thick enough that it has enough mass to have its own gravity?
When the ring rotates at orbital speed, there is no perceived gravity on either side. If the ring rotates faster than that, one would perceive a force pulling them away from the central star. If, however, the ring rotates below orbital speed its mass would be pulled toward the central star (along with anyone standing on its outer surface). The forces pulling it towards the star would compress its structure (the opposite of the traction forces a faster ring with positive gravity on the inner surface would experience).
I think (I haven't checked the numbers) the ring's mass can't be neglected, but, because of the shape, it'd act as an extra mass within the central star. The ring is, in fact, in its own orbit.
Sunlight could still be reflected by a fleet of orbiting mirrors that wouldn't orbit the star in sync with the ring and thus would not be subjected to the structural forces of the ring. In case the ring is built around a smaller star with a spectral emission very different from the one preferred by the ring builders, the mirrors could adjust that too.