As we know, Sauron poured himself into the Ring:
"-but secretly Sauron made One Ring to rule all the others --- And much of the strength and will of Sauron passed into the that One Ring-" (Silmarillion)
Sauron and the Ring are naught but pieces of a whole; without the other, the other's incomplete. And they're trying to become whole.
Actually, a while back, I happened to discuss the Ring a bit more in depth IRL.
During that discussion, one of us 'simplified' the relationship of the Ring & Sauron: Being made out of himself, the Ring is 50% Sauron. Sauron without the Ring is 50% Sauron. But Sauron with the Ring, designed to enhance and empower him, is 150% Sauron.
[center]Jestem tym, czym jestem.[/center]