What powers of attraction does a Ring have or need?

LOTR, Book I, Ch. 2 The Shadow of the Past:

‘A Ring of Power looks after itself, Frodo... It may slip off treacherously, but its keeper never abandons it. At most he plays with the idea of handing it on to someone else’s care – and that only at an early stage, when it first begins to grip. But as far as I know Bilbo alone in history has ever gone beyond playing, and really done it. He needed all my help, too. And even so he would never have just forsaken it, or cast it aside. It was not Gollum, Frodo, but the Ring itself that decided things. The Ring left him.’

It seems implied that this is one of the Rings' specific attributes. But does it need to be?

If I found a ring tomorrow that made me invisible, nevermind a Ring of Power, I highly doubt I would ever 'forsake or abandon it,' leading me to wonder how much of this property an inherent attribute of a Ring and how much is merely a byproduct of the powers it bestows?