Heart Ring Location - Morrowind
Heart Ring is obtained towards the end of the final quest of the game, The Citadels of the Sixth House. After you beat Dagoth Ur in your first fight with him in the Facility Cavern area he'll drop the Heart Ring on the ground as he flees to the adjacent room. You'll find this ring right in front of the door that leads you to the Akulakhan's Chamber.
Despite the fact that this ring is meant to be obtained during the final quest of the game - you're actually able to come here at any point during the game and acquire the ring. I've personally tested and confirmed this in the video I made about the ring.