Can an asteroid have rings? I know an asteroid can have moons, but how about rings? Does it also get affected by the Roche limit?

The asteroid Florence has two moons.
The simple answer is yes, an asteroid can have rings. The current known example is 10199 Chariklo, whose rings were discovered in 2014 (see Braga-Ribas et al. (2014)). It actually has two rings, at distances of 391 km and 405 km, 7 km and 3km in width. As the discovery paper notes, they may be the (young) remnants of a debris disk around Chariklo, and have some moderately-sized bodies embedded in them. The observations indicated that water ice is likely present. Another small body, 2060 Chiron, may also have rings (see Ortiz et al. (2015)). Again, water ice is thought to be a major component of the putative rings.

Both ring systems were hypothesized in part due to evidence from stellar occultations, i.e. cases where the minor body passed in front of a star. Stellar occultations have proved useful in many cases involving small bodies, such as Kuiper Belt Objects.

As Ortiz et al. noted, Chariklo and Chiron are the two most massive members of a class of minor planets called centaurs. They orbit (unstable) in between the giant planets of the Solar System. Insights into ring formation around centaurs could perhaps provide information about centaur formation and evolution. It seems likely that the rings form from either


The impact of a smaller body on the surface of the centaur
The breakup of a satellite orbiting the centaur (as it crosses the Roche limit, as you speculated)


Now, rings have not been observed around Trans-Neptunian Objects in similar stellar occultation data, leading the authors to hypothesize that centaurs are a separate population. More observations could shed some light on this hypothesis.

A final interesting point I'd like to make is that many moons around minor planets have been found, but only one or two ring systems are known.