Quote:
Does Gollum associate himself with the Ring in some strange way, to the point that in his mind they are really the same thing?
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The personality of Gollum seemed to be wholy based on the possession of the Ring. So, as Oddwen pointed out with the quote, with the loss of the Ring came something of a loss of self. The Gollum personality didn't have anything to attach to and so it was "lost", so to speak.
Smeagol, on the other hand, was the initial personality, and so wasn't "lost". So came the Smeagol/Gollum arguments. The Smeagol personality was able to voice his opinion (heh) because of the "loss" of strength in the Gollum personality. Unless I'm much mistaken, Gollum and Smeagol didn't argue much (or at all) in The Hobbit, when the Ring was surmised to be safe in his/their hut.
Abedithon le,
~ Saphy ~