I googled this. There is a Roche limit around planets that determines whether or not you get rings vs moons. If you have debris within the Roche limit, the tidal forces of the planet keep the debris from coalescing (the tidal forces exceed the force of gravity). Outside the Roche, you get moons. The Earth's Roche limit is 10,000 km, so based solely on that, we could have stable rings between the Earth`s surface (6,400 km) and the limit. However, other factors affect the formation of rings, so those might play a role (e.g., melting of icy debris too close to the sun, the effect of gravity of any nearby moons, the density of a planet, among others). This is googled info, so take it with a grain of salt.
I'm curious to know why rings around smaller planets would only last 2000-3000 years.