Interesting question. A ring is often formed around a celestial body when its gravity rips apart another smaller celestial body. The Sun is really huge, so it could destroy any object that is not dense enough. Just Google about the Roche Limit for more informations (and better explanations).
Now, take a look at our solar system : You have the asteroid belt between Mars and Jupiter, and you also have the Kuiper belt beyond Neptune. Would you consider these being rings ? They sure are not as smooth as Saturn's, but to my mind, they still are rings.So yes a star can have a ring system.