The Ring works by desire

[The Saucepan Man]

it's often stated the Ring worked by letting the bearer think he could control it's power or by clinging on the greatest power of the wearer.
I personally don't think this is the case,but rather that the Ring worked by what the bearer desired.

I don't think that these two statements are contradictory. Where the bearer or anyone else that it sought to corrupt desired power, then it tried to delude that person into thinking that they could achieve that power by mastering it. As Willie said:

The ring will let the bearer think that the bearer can use it to achieve its wants and desires.

I also agree with Willie on this:

Well, it worked both ways. They are both trickery and deceit. And they are both linked together. You can't really say the ring has just one aspect to how it works because there are many.

There is no one particular way in which the Ring worked in order to achieve what it desired, and what it desired was to return to its Master. As I see it, the Ring was able to influence the thoughts of those that came into close contact with it, and it influenced them in whatever way it perceived would get it closer to its Master.

So, it appealed to Smeagol by making him regard it as his by rights, his birthday present, presumably because it perceived that he was more subsceptible to its influence than Deagol, and possibly more useful because of his sly and secretive ways. Then, when it felt its Master's power rising and perceived that Gollum was no longer of any use to it, it abandoned him no doubt expecting to be found by a Goblin, rather than the "most unlikely person imaginable" (clearly a mistake on its part).

It seems that, despite its success in persuading Frodo to put it on at the most inopportune moments (The Prancing Pony, Weathertop), it was having difficulty breaking his Hobbit spirit. So, it went to work on another member of the Fellowship, choosing the most susceptible member, and (save for Aragorn) the one likely to get it closest to Mordor - Boromir. And it worked by appealing to his desire to use it for the good of his people, and by playing on his pride and ignorance in thinking that he would be able to use it in this way. (I mean ignorance here in the sense of Boromir ignoring the warnings of the wise concerning the dangers of trying to use it. He thought that he knew better.)

It also tried to tempt Galadriel, which is interesting. It appealed to her desire to remain in ME and preserve her realm against the irrevocable onset of Man and the change that Man represented. It probably thought that she would attempt to use it to overthrow Sauron and that she would probably fail in the attempt. And, in my view, it was probably right. Galadriel, to her great credit, was able to resist its temptation and was rewarded in this by being allowed to return to Valinor.

When there was no other option, the Ring also tried to tempt Sam. It appealed first to his gardening instincts, presumably thinking that someone trying to cultivate the Plateau of Gorgoroth would be easy prey for Orcs [img]smilies/wink.gif[/img] .Then, when that it didn't work, it played on his sneaking desire to be a hero which, again, would have led him straight into trouble with Orcs and goodness knows what else. To Sam's even greater credit, he too resisted its temptation and, ironically, was later perceived by the Orcs to be a great Elven Hero.

Of course, ultimately, the Ring paid the price of having corrupted Gollum, for he was so eaten with desire for it that he tried to seize it at the most opportune (for the good guys) of moments. Interestingly, it seems not to have picked up on its Master's own strange disquiet over Gollum, mentioned in The Hunt for the Ring in UT (which foreshadowed his ultimate doom). Or perhaps it did share that disquiet and this is the reason why it chose to abandon him.

By the way, I make no apologies for treating the Ring here as another character, for that it precisely what it is. Gandalf regarded it in the same way in A Shadow of the Past. And it was, after all, a part of Sauron, and so is just as much a character in the story as him.

Finally, on invisibility, I do not think that this was a power that it conferred just on those of its bearers that desired it. I see it as an inherent power of the Ring, linked to its connection to the spirit world. The wearer does not become truly invisible, but passes into the spirit world when wearing it. Hence, Frodo is able to see the Nazgul, creatures of the spirit world with no corporeal bodies, in their true forms while wearing it. Tom Bombadil excepted, I think that most, if not all, of those who wore it would have become invisible, although some, such as the Istari and those Elves (such as Glorfindel) who had "dwelt in the Blessed Realm" and already partly resided in the spirit world, would probably have been able to control this ability.