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:
Isildur himself escaped by means of the Ring, for when he wore it he was invisible to all eyes; but the Orcs hunted him by scent and slot, until he came to the River and plunged in. There the Ring betrayed him and avenged its maker, for it slipped from his finger as he swam, and it was lost in the water. Then the Orcs saw him as he laboured in the stream, and they shot him with many arrows, and that was his end.
This obviously attributes the betrayal and vengeance to the Ring.
The passage describing Isildur's death in the Unfinished Tales is similar:
There suddenly he knew that the Ring had gone. By chance, or chance well used, it had left his hand and gone where he could never hope to find it again.
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:
"A Ring of Power looks after itself, Frodo. It may slip off treacherously, but its keeper never abandons it. ... It was not Gollum, Frodo, but the Ring itself that decided things. The Ring left him."
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.