My Precious?

[Firefoot]

Recently, I was rereading The Hobbit, or, more specifically, the "Riddles in the Dark" chapter, when this occurred to me. Obviously, the Ring had long since consumed Gollum's mind, and it was the most valuable and precious thing he had. The question is, what is really Gollum's precious? Let me explain:

The first time we hear Gollum speak is immediately after he is described to us:

"Bless us and splash us, my precioussss! I guess it's a choice feast; at least a tasty morsel it'd make us, gollum!" And when he said gollum he made a horrible swallowing noise in his throat. That is how he got his name, though he always called himself "my precious." (italics Tolkien's, bolding mine.)

And a few sentences later:

What iss he, my preciouss?" whispered Gollum (who always spoke to himself through never having anyone else to speak to).

It seems pretty clear that here, "my precious" is in fact Gollum himself. In LotR, it seems very explicit that "my precious" is the Ring. This is how Frodo and Sam understand it: Sam says in "The Stairs of Cirith Ungol": "But I was forgetting - his Precious. I suppose the whole time it has been The Precious for poor Smeagol." When Frodo talks to Gollum about the Ring, he always uses the term "precious." With the possible exception of in "The Taming of Smeagol", there would not seem to be a doubt that Gollum uses "my precious to refer to himself - even he calls himself Smeagol. This might be passed off as a difference or controversy between the two books, except towards the end of "Riddles in the Dark", there is this:

He had a ring, a golden ring, a very precious ring. (...) "Where iss it? Where iss it?" Bilbo heard him crying. "Lost it is, my precious, lost, lost! Curse us and crush us, my precious is lost!"

Usually in the Hobbit, the ring is "his birthday present", but here, the Ring is certainly the precious being referred to. And, it LotR, he appears to be using "the precious" in both ways within sentences of each other:

"Ach, sss! Cautious, my precious! More haste less speed. We musstn't rissk our neck, musst we, precious? No, precious - gollum!" He lifted his head again, blinked at the moon, and quickly shut his eyes. "We hate it," he hissed. "Nassty, nassty shivery light it is - sss - it spies on us, precious - it hurts our eyes."
He was getting lower now and the hisses became sharper and clearer. "Where iss it, where iss it: my Precious, my Precious? It's ours, it is, and we wants it. The thieves, the thieves, the filthy little thieves. Where are they with my Precious?"

So the question is: what is "the precious", really? Does Gollum associate himself with the Ring in some strange way, to the point that in his mind they are really the same thing? He is certainly attached the Ring strongly enough. If this is so, what does Gollum mean when he swears to serve the master of the precious "by the precious"? Is he really swearing by the Ring, as Sam and Frodo think? Or does Gollum simply use the same term for the two things that are most precious to him: the Ring, and his life?