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Free Rmodules... 
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#1
Jan1313, 01:35 PM

P: 247

From my textbook:
A free Rmodule is "A left Rmodule F is called a free left Rmodule if F is isomorphic to a direct sum of copies of R..." I know that another definition of an Rmodule 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 


#2
Jan2113, 02:51 AM

P: 428

[itex][/itex]
$$F\cong\prod_{\alpha\in J}R_\alpha$$ 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). So that seems to suggest that a left Rmodule does indeed have a basis. Now let's consider if we think a left Rmodule with a basis is a free module. Uh, never mind, I'll leave that for someone else 


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