Here it is, Mr. Zetsu, I found it. Maybe now I can become a member of the Akatsuki - Tobi One explanation for the rings disappearance could be attributed to the Akatsuki members defeat over time.

However, behind the scenes, the true puppet master was Tobi, also known as Obito Uchiha. While masquerading as Madara Uchiha, Obito used the Akatsuki to execute the “Eye of the Moon Plan” that ...