I think Frodo is also influenced by Bilbo's offer to take the ring. Bilbo, aged 129, resignedly says "Bilbo the silly hobbit started this and he must finish it." Perhaps he also was motivated by desire to possess the ring again but I think his entirely serious offer to undertake the task confirmed for Frodo the idea that it was his duty and his alone to destroy the ring. He was Bilbo's heir, he had inherited the ring, it was his to destroy. And so he was motivated by both a sense of duty and by desire for the ring.
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