J.R.R. Tolkien> The Lord of the Rings / The Silmarillion

His words were proud, and all eyes looked upon the ring; for he held it now aloft, and the green jewels gleamed there that the Noldor had devised in Valinor. For this ring was like to twin serpents, and their heads met beneath a crown of golden flowers, that the one upheld and the other devoured; that was the badge of Finarfin and his house.

-J.R.R. Tolkien
The Silmarillion

The Ring of Barahir was a gift to Barahir and his descendents by Finrod after Barahir saved his life in the Dagor Bagollach. The ring was not was of the Rings of Power. It was made in Valinor by the Noldor and was given with a pledge to aid Barahir and his kin whenever it was needed. When Barahir was slain in Dorthonion, the hand that bore the ring was cut of as proof of his death. Later, his son Beren recovered both hand and ring.
But when the wolf came for Beren, Felagund put forth all his power, and burst his bonds; and he wrestled with the werewolf, and slew it with his hands and teeth; yet he himself was wounded to the death....Thus King Finrod Felagund, fairest and most beloved of the house of Finwë, redeemed his oath; but Beren mourned beside him in despair.

-J.R.R. Tolkien
The Silmarillion

During the quest of the Silmarils, Beren brought the ring to Finrod, who fulfilled his pledge and died saving Beren in the dungeons of Minas Tirith (Not the Minas Tirith of Gondor). It is not certain exactly how the ring survived the reminder of the First Age, but it apparently entered into the possession of the Faithful of Númenor during the Second Age. During the Third Age it was one of the heirlooms of the House of Isildur. After the fall of Arthedain it was kept at Rivendell, along with other relics of the North-kingdom.

The ring was made of a band shaped like two serpents, one eating the other and supporting a crown of gold flowers.

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