Ok, so it's not a brown dwarf. Regardless, it puts out far more energy than it receives from solar input, so the hypothesis remains more or less sound. Jupiter likely doesn't have a ring system of any complexity because of amount of energy output preventing ice crystals from collecting at an altitude where rings can form and lack of shepherd satellites in those same altitudes to organize and form the rings.
I doooon't think that's how it works. The ring system for Jupiter isn't very complex because of a lack of a huge amount of icy source material being shed to produce and maintain a vibrant ring system, plus there's a huge amount of radiation which whittles away fine particulate matter as well.
That said, Jupiter's ring system (while certainly not as complex as Saturn's) isn't exactly mediocre. You'll notice that there are moons associated with all of them as well.


Edit: I notice you mention ice over dust, which begs me to ask where the ice particles would be coming from in the first place? All of Jupiter's inner moons are dusty/rocky instead of icy. Europa would be the best option, and orbital resonance with the other Galilean moons would prevent a ring from forming there. So it's not that heat is being given off by Jupiter to prevent ice from accumulating, it's that there's no ice there to begin with.