
#1
Jan1313, 01:35 PM

P: 245

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 


Register to reply 
Related Discussions  
Finitely generated modules as free modules  Linear & Abstract Algebra  10  
Vector spaces as quotients of free modules  Calculus & Beyond Homework  8  
Torsionfree modules over a Discrete Valuation Ring  Linear & Abstract Algebra  1  
torsionfree modules  Linear & Abstract Algebra  1  
free modules  Calculus & Beyond Homework  1 