A thin Dyson ring is entirely possible.. Imagine Saturn's rings as a set of nested rings like the concentric grooves on a DVD. Now select one of those nested rings , keep it in orbit around Saturn, and throw the rest of them away. Replace that ring, which is a circular string of unconnected particles, with a chain (which is a string of interlinked "particles" that happen to be chain links). Make the links the size of a large building so people can live and work in them, and put solar panels on the sun-facing sides. Add springs to allow a bit of flex, since the orbit probably won't be perfectly circular, and the ring is essentially finished.
Imagine what happens if the orbit becomes elongated. The speed of the chain links will increase to a maximum at their closest approach to the sun, so the chain will need to stretch. At the maximum distance from the sun, the chain will need to compress. That can be taken care of by making the chain links long enough and loose enough.
If the orbit were perfectly circular, and the ring were solid instead of loosely linked like a chain, there would be some other forces to consider. The inside of the ring, closest to the sun, would be moving a bit too slowly for its centripetal acceleration to balance the sun's gravitational acceleration. The outside of the ring would be moving a bit too rapidly. This would result in tidal stresses. Tidal stresses approach zero as the thickness of the ring approaches zero, so the tidal stresses should be easily manageable if the ring is no more than a few hundred meters thick. (We know that the ISS doesn't get torn apart by tidal forces, so it"s not necessary to do the math for that.)
However, a perfectly circular orbit would be very difficult to obtain.
The question was about a perfectly rigid ring. Perfect rigidity results in infinite forces in a lot of situations, and I suspect a perfectly rigid Dyson ring would not be an exception. For example, the pressure at the point of contact between two perfectly rigid spheres that collide would be infinite regardless of the collision speed or the masses of the spheres. That means there can't be such a thing as a perfectly rigid sphere (or ring, for the same reason) unless it is made of infinitely strong material. That's too many infinities for me to swallow!
If the "perfectly rigid" requirement is taken away, the question can be approached sensibly from another direction. We know for sure that we can build a stiff ring around a bowling ball, which is a gravitating object. Scale up both the bowling ball and the ring until the compression in the ring approaches the limit for, say, titanium. Now rotate the ring at orbital velocity to relieve the compression, and scale up some more, taking perturbations into account. Stop before the tidal stresses get close to the limits for titanium. That's your Dyson ring, and it means that as long as perfect rigidity is not required, there is a solution.
