Orbital mechanics, has something to say about stable orbits. Material closer to planet must travel faster, material further away must travel slower. A ring of any sizeable width will swirl over time, spinning faster or slower depending on distance from the body it's orbiting.

Not an astrophysicist, but I believe that's the same driving force that can make rings. When a moon's orbit degrades to the point where the difference in gravity between the near-side and far-side is greater than the moons own binding gravity, the near side will naturally accelerate away from the moon while the far-side luffs off behind.

So no, a ring cannot be a solid plate. But a series of hoops with bearings between them? Well probably not that either since rings aren't flat but undulate up and down in the plane of their orbit due to... Science.