What if??

[Farael]

I don't think Smaug's fire would have been hot enough to destroy the One Ring (if any Dragon's fire ever could have).

I don't have time now but remind me to check tonight. I'm almost sure that in The Fellowship of The Ring, either when Gandalf throws the Ring to Frodo's fire or when, at Elrond's council they talk about destroying it, someone (probably Gandalf) says that the fire of the dragons might have been able to, but there are none left.