What is the definition of a dollar now without gold backing? It seems that a dollar is now just an entry in a book somewhere in some bank. Or it is the federal reserve note? Or is it both? I understand the concept of the monetary base. But what I don't understand is how this works. If a dollar is in part just an entry in a book, what stops some bank from simply claiming they have an infinite amount of dollars? Or is there some central registry that claims to keep track of how many dollars each bank has? (This would seem absurd to me. How then would foreign banks holding US dollars keep track of how many dollars they had?) If there is no central registry, then how do you prevent people claiming to have dollars on their books?
I think explaining what a dollar is would go toward answering these questions.