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Free R-modules

  1. Jan 13, 2013 #1
    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
    Last edited: Jan 13, 2013
  2. jcsd
  3. Jan 21, 2013 #2
    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).

    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

    The product above consists of copies of R.
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