Yes,
Tolkien clearly wanted to fit his rings into the the tradition of "
magic rings"; or to put it the other way round, to draw the whole world of fairy-tale and Romance literature into his own legendarium -- including rings.
But I want to suggest a further thought. LOTR can be read in many ways: as a pure adventure story, as a series of cautionary
tales, as tragedy, comedy, irony. One way, especially in
Fellowship, is as a series of unfolding mysteries, making it, for
Frodo, what we'd call a "
mystery story".
But that is a relatively new genre, as is, from Tolkien's perspective, the use of "mystery" itself, in that sense, I can't recall if he used the word in his story that way -- maybe someone can give an answer -- but it traditionally had religious associations, rather than with a puzzle to be solved.
He preferred a term used widely in the early Medieval world he was most familiar with and comfortable in: "
riddle", which first features in
The Hobbit, but which appears again and again throughout LOTR. Some of the finest lyrics in Old English are
riddles, and I think the idea would occur to him naturally.
How does a riddle work? Northop Frye puts it this way:
"Circle of words" is certainly suggestive, as is "containment";
Sauron clearly intended to "contain" the other rings, but there are other implications, which I won't go into here, beyond pointing out that the rest of
the Ring poem postdated the inscription on
the One, making "them" in its verses at least ambiguous.
'How then do you read this riddle?' asks
Gimli, and besides showing that Tolkien the philologist was well aware that "riddle" was originally the cognate object of "read", brings us round to
Balrog's question again: why put the words on the Ring? Various answers have been given, and I'll just add this, in relation to the aspect I've been talking about: the Ring inscription is normally invisible -- it
cannot be read -- making it a "riddle" indeed.
