All rings and amulets (except chargeable rings that are at +0 charge, and the meat ring) cause one additional nutrition point to be deducted once every twenty turns. This is additive -- a player wearing two rings and an amulet will lose three additional hunger points every twenty turns. This sense of "ring hunger" is separate from the hunger-causing effects of the ring of regeneration, ring of conflict, and the ring of hunger, all of which cause a nutrition point to be lost every other turn. The ring of slow digestion prevents ordinary nutrition consumption, but still causes ring hunger (if it didn't, it would be a ring of no digestion.)
Dylan O'Donnell and Kevin Hugo have a 3.4.3 spoiler that more or less remains current for NetHack 3.6.1: food spoiler.