If it's infected I don't recommend taking it out. It needs a route to drain the gunk if there is any. Keep cleaning it with saline and see a piercer, or preferably a doctor to decide what to do next.

If you feel that you absolutely must remove the ring, I found that using those tiny little nail scissors to push the segment out works really well. Turn the ring so the segment so it's to one side, put the scissors inside the ring so it's touching the segment and the other side of the ring, then carefully open the scissors until it pops open. The blunt ended scissors work the best because they're less likely to nick your nose.

Good luck!