The Inner Ring

I don't know if C.S. Lewis' essay The Inner Ring has any direct connection to Tolkien's epic masterpiece The Lord of the Rings.  However, it would be hard to read this brilliant C.S. Lewis essay without considering the possibility that it might.  Tolkien wrote of "One Ring to rule them all,/ One Ring to find them,/ One Ring to bring them all,/ And in the darkness bind them."  It is at least conceivable that Tolkien's "One Ring" and C.S. Lewis' "Inner Ring" actually refer to the same thing.