Debris ring formed from breakup of extreme-high rotation rate of rigid ring

In Larry Niven's Ringworld, where there is a rigid rotating ring with a radius of 1AU with the Sun at the centre, he introduces a fictional material "scrith", which apparently is strong enough to resist the mechanical stresses around the order of 10^16 pascals induced by the high rate of rotation to induce a centrifugal acceleration of 9.81 to maintain Earth-like artificial gravity. Realistically, no material currently exists that can withstand that amount of stress and the ring would break up.

After extensive research the only things I could find that was barely relevant was the calculation of the Roche Limit, however my case is breakup due to self-rotation instead of tidal forces between the primary body and itself. How would the ring break up into? What are the parameters of this new "debris" ring after each fragment of the original ring has occupied new orbits around the Sun? As in, how does one calculate the outer and inner bound of the rings, what would be its thickness and density?

The ring velocity needed to maintain the standard gravity is $\sqrt{g R}\approx 1200 \,\text{km}/\text{s}$ (actually slightly larger since the Sun's gravity has to be compensated). That is more than 25 times the velocity needed to leave the Solar system. So, there would be no 'debris ring', all the fragments leave the Solar system. Within 50 days they would be beyond Neptune orbit, and continue flying away indefinitely.