There are actually several themes assigned to the Ring. The main Ring theme is called "the History of The One Ring" and will be fimiliar for playing over the Lord of the Rings title cards. There are hints of it under The Hobbit title card as well.
It's alluded to in the "Smoke Rings" theme as it appears in the Hobbit. In fact, the Ring theme is a component in the very theme that opens the first Hobbit film, which blends the Ring theme with the House of Durin theme. The History of The One Ring is the closest the series has to a main theme.
But, much like Wagner, Howard Shore wrote several other themes assigned to the Ring, all distinct but nevertheless related musically. There is a theme for "the Seduction of the Ring", usually sung by a Boy Choir. There is a theme for the ultimate "Fate of the Ring" (heard in the Two Towers) that transforms into yet another (distinct) theme for "The Destruction of the Ring" and of Mordor.
Also, several of the Mordor themes are linked to the Ring's themes and can be seen as relating to the Ring as well. "Sauron's theme" (or the Mordor theme) and - by proxy - "The Necromancer theme" from the Hobbit, are often referred to as the theme for "the Evil of the Ring". This theme is gradually intervowen with the History of the Ring theme and once with the Seduction theme, leading up to a "meta" theme that combines all three themes at Mount Doom.
"The Power of Mordor" theme (related to the Ringwraiths/Servants of Sauron theme) has also been described by Howard Shore once as the theme for the Power of the Ring. For obvious reasons, "Smeagol's theme", "Nameless Fear" theme, Smaug's main theme (as well as his ascending theme and demise theme) and "Evil Times" theme (and therefore the related "Death theme" from the Hobbit) are also closely related to the Ring theme. Hell, there is even a connection to the Shire theme.
There are even recurring musical ideas not yet cannonised as themes that seem to relate to the Ring. There is a choral section that reappears through the score, reading as "Tercano Nuruva" (the herald of death) that seems related to the Ring.
If this makes you dizzy there are a lot more themes where this came from. There are over one hundred themes in the Lord of the Rings alone, and easily fifty or sixty more in the Hobbit.