The One Ring gave powers to its wearer according to their stature. It gave invisibility and immortality of a kind to mortal beings, the ability to dominate minds to those strong of will, and would greatly enhanced the existing power of those with power (Sauron, Gandalf and Galadrial, for example). The powers granted varied, as Sauron was not rendered invisible when he was wearing the Ring in the Third Age battle against Elendil and Gil-Galad. Hence invisibility itself was not inherent to the Ring, but dependant on the wearer.
However it also works its power through the mind of its bearer, and also by affecting those in its immediate vicinity (Smeagol murdering Deagol, Boromir attacking Frodo). It does not exert any influence of inanimate or unthinking objects, so the chain for instance would not become invisible.