Free R-Module Basics: What is a Copy of R?

[Artusartos]

From my textbook:

A free R-module is "A left R-module F is called a free left R-module if F is isomorphic to a direct sum of copies of R..."

I know that another definition of an R-module a module with a basis...but I don't know how to connect that definition with this one. Also, what does "copies of R" mean?

Thanks in advance

 

[algebrat]

From my textbook:

A free R-module is "A left R-module F is called a free left R-module if F is isomorphic to a direct sum of copies of R..."

Can we think of it as
$$F\cong\prod_{\alpha\in J}R_\alpha$$
This is the underlying abelian group (analogous to vectors in vector space), and it looks like there is a natural way to multiply on the left by elements of R (analogous to scalars in a vector space).

I know that another definition of an R-module a module with a basis...but I don't know how to connect that definition with this one.

For F above, a basis could be elements like (1,0,...,0), (0,1,0,...,0) and so on. Notice each coordinate alone looks like R.

So that seems to suggest that a left R-module does indeed have a basis. Now let's consider if we think a left R-module with a basis is a free module. Uh, never mind, I'll leave that for someone else

Also, what does "copies of R" mean?

The product above consists of copies of R.