My Math Forum Does "free" mean "finitely generated"?
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 July 13th, 2017, 11:29 PM #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 13th, 2017, 11:37 PM #2 Senior Member   Joined: Oct 2009 Posts: 367 Thanks: 122 No, it doesn't. There is no relation between the two. https://en.wikipedia.org/wiki/Free_object Thanks from zzzhhh
 July 13th, 2017, 11:39 PM #3 Senior Member   Joined: Oct 2009 Posts: 367 Thanks: 122 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. Thanks from Maschke, Country Boy and zzzhhh
 July 14th, 2017, 10:23 AM #4 Global Moderator   Joined: Dec 2006 Posts: 18,847 Thanks: 1568 Did you mean "pointwise"? How are free generators defined?
July 14th, 2017, 10:33 AM   #5
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Quote:
 Originally Posted by skipjack Did you mean "pointwise"? How are free generators defined?
Where did you read "free generators"?

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