Frodo didn't "allow" himself to be consumed.
The Ring corrupted everyone who came in contact with it.
Frodo lasted much longer than anyone else.
Gollum became a murderer after a glimpse of the Ring. Boromir was consumed just by the knowledge that it existed. The mighty Isildur fell for it immediately, ignoring the advice that it should be destroyed. Saruman was twisted by the thought of possessing it. Gandalf and Galadriel both feared what it might do to them. Even down-to-earth Sam was tempted by it.
It was inevitable that Frodo would have his life destroyed by the Ring. He could only find peace in The West.
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