From Galadriel in Lord of the Rings Fellowship of the Ring
It betrayed Isildur to his death
Why does Galadriel say it's a betrayal by the One Ring? Wasn't it Isildur who let it fall?
How could a ring betray someone?
It is said that the ring has a will of its own, or rather it has a bit of the will of Sauron. The ring intentionally slipped off Isildur's finger; Isildur did not merely lose his grasp.
If I recall correctly, Isildur tried to use the ring to escape from a group of orcs. He put it on, became invisible and tried to swim across a river. But in the middle of the river, the ring failed him and slipped off and he became visible, so the orcs shot him, and he died. Hence, the ring betrayed him.
And the ring remained in the river until it was found many years later and ended up in Gollum's (Smeagol's) possession.
Galadrial is paraphrasing Tolkien himself in the Silmarillion:
This obviously attributes the betrayal and vengeance to the Ring.
The passage describing Isildur's death in the Unfinished Tales is similar:
Again, it can be read as attributing the slipping off of Isildur's finger to the Ring itself.
Finally, Gandalf definitely believes the Ring has this ability, as per the Fellowship of the Ring:
The belief of the Wise in Lord of the Rings is obviously that the One Ring is sentient enough to slip off its bearer's finger, hence Galadrial's comment.