First, this urban legend should be put to rest.

J.R.R. Tolkien did not write "The Lord of the Rings". First, the proper title is The Red Book of Westmarch. The section you are referring to is called "Of the War of the Ring and the Return of the King". He simply translated it. Just because the race of Hobbits is extinct does not mean we should ignore their writings or pass it off as done by Men.

It was written by the hobbits Bilbo and Frodo Baggins (the Finding of the Ring and the Battle of the Lonely Mountain of the former, and the War of the Ring for the latter). Additional parts were written about Arnor, Gondor and Rohan by Peregrin Took and Meriadoc Brandybuck.

Ringworld was published in 1970. The Hobbit was published in 1937. Lord of the Rings was published in 1954-55.

It is therefore highly unlikely that Tolkien was influenced by Niven's novel.