Other magic Rings?

[Sardy]

Forgive me if this is something unique to the films. I've read the Trilogy, Sil, Hobbit, and HoME several times and can't recall whether Gandalf's line "There are many magic rings in this world, Frodo Baggins, and none of them should be taken lightly" was unique to the films or if it also appeared in the books.

But either way, the question remains: What other sorts of magic rings (or other artifacts) existed in ME, and where did they come from? And how common were they that Gandalf, despite his words to Frodo, considered them "harmless" enough (or at least unpresuming enough) that he didn't immediately jump to the conclusion that Bilbo's was a ring of power?

I'm more curious about the "every day" magical items in ME, as opposed to the more unique and powerful items (Galadriel's mirror, the staffs of the wizards, the morgul blades, etc.)