Symbolism, Imagery, Allegory
Given the novel's emphasis on cycles, it's not super surprising that rings (which are circles, of course) carry a lot of symbolic weight. There's also the subplot where Bernard has to travel to Mordor in order to burn the One Ring in Mt. Doom… oh, wait. Wrong book.
While yearning for the ability to use language to draw connections between seemingly disparate items, Louis expresses a desire to forge "a steel ring of clear poetry that shall connect the gulls and the women with bad teeth, the church spire and the bobbing billycock hats" (4b.41). For Louis language is kind of like a fancy key ring that can bring even radically different items (or, in this case, people) together.
This is not the only instance in which the image of a ring is used to signify connection. For example, as the six narrators and Percival are about to part ways before Percival leaves for India, Louis notes that a "circle in [their] blood," which breaks often due to their differences from each other, closes again and "[S]omething is made" (4b.77). Psst: that something is a ring.
The fact that rings are circular is also important. As you may have gathered, the novel is kind of obsessed with cycles. The chapter intros, which track the progress of the sun over the course of 24-hour day, serve as the first (and most glaring) clue to this preoccupation.
The characters sometimes cycle through a series of emotions or opinions before returning to the original one. Bernard, for one, goes from feeling that things are "good" to experiencing intense despair… and then goes back to feeling good again. So, as circles, rings do double duty, serving as a symbol of both interpersonal connection and the cyclical nature of many life activities (and emotional rollercoasters).
Of course, there are many other rings—including literal ones—mentioned in the book, including jewelry that the characters wear (like Miss Lambert's amethyst ring). Rings, after all, are widely symbolic of prestige (you need cash in order to buy diamonds) and landmark life events (wedding rings, class rings, and teething rings).