Could someone explain me what the word "copies" in Terms of Rings/Vectorspaces/Modules means?
E.g in the context "For a ring R, the smallest subring containing 1 is called the characteristic subring of R. It can be obtained by adding copies of 1 and −1 together many times in any mixture" or
"A free R-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces."
Do copies have special properties? How can one think about them and why is it usefull or easier to work with them instead of the whole ring?