Ouch! I don't really know much about the topic, but I know that when my molar was removed it fell apart into 3 pieces in the process, at which the dentist exclaimed that he'd never seen such a thing
All I do know is that having a cavity can increase your chances of getting more, and if the tooth is already damaged, it's probably going to hurt alot as time goes on.
I'd talk to your dentist though and make your own decision after getting their opinion. I'd guess that if there's no gum disease or root damage, it's not going to be coming out on its own anytime soon.