The accepted answer is correct, and as is the answer from Jason Hutchinson, and together they answer why no more rings were created while the One existed.
After the One Ring was destroyed, all Rings of Power ceased functioning as magic rings. It is likely any such rings made after that point would not have worked as magic rings; the specific workings aren't explained (they're magic!) but one may think of it as the One Ring having some special power source all Rings used, like it was the source of electricity for machines, and any new rings made could not use the same techniques to produce the effects.
This is glimpsed, I think, by the loremaster Saruman in Return of the King:
‘I did not spend long study on these matters for naught. You have doomed yourselves, and you know it. And it will afford me some comfort as I wander to think that you pulled down your own house when you destroyed mine.
He is speaking of the decline of the Elves, which was staved off by the Elven Rings, but he seems also certain that no method for the same kind of method for evading this doom will work any longer, which implicitly includes making more rings.