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July 14th, 2017, 12:29 AM  #1 
Newbie Joined: Jun 2017 From: Earth Posts: 17 Thanks: 0  Does "free" mean "finitely generated"?
I often hear people say "free XXX" in which XXX is some algebraic structure like abelian group or module. Does "free" mean "finitely generated" (the set of generators is finite) in these context? Thanks.

July 14th, 2017, 12:37 AM  #2 
Senior Member Joined: Oct 2009 Posts: 142 Thanks: 60 
No, it doesn't. There is no relation between the two. https://en.wikipedia.org/wiki/Free_object 
July 14th, 2017, 12:39 AM  #3 
Senior Member Joined: Oct 2009 Posts: 142 Thanks: 60 
For example, $\mathbb{R}^\mathbb{N}$ with usual pointswise operations is an $\mathbb{R}$module that is free but not finitely generated. On the other hand, $\mathbb{Z}_2$ is a $\mathbb{Z}$module that is finitely generated but not free. 
July 14th, 2017, 11:23 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,148 Thanks: 1418 
Did you mean "pointwise"? How are free generators defined?

July 14th, 2017, 11:33 AM  #5 
Senior Member Joined: Oct 2009 Posts: 142 Thanks: 60  

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