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July 13th, 2017, 11:29 PM   #1
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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.
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July 13th, 2017, 11:37 PM   #2
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No, it doesn't. There is no relation between the two.
https://en.wikipedia.org/wiki/Free_object
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July 13th, 2017, 11:39 PM   #3
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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.
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July 14th, 2017, 10:23 AM   #4
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Did you mean "pointwise"? How are free generators defined?
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July 14th, 2017, 10:33 AM   #5
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Quote:
Originally Posted by skipjack View Post
Did you mean "pointwise"? How are free generators defined?
Where did you read "free generators"?
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