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Topic: Basis of module and basis of vector space qs.
Replies: 9   Last Post: Nov 21, 2013 10:08 PM

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magidin@math.berkeley.edu

Posts: 11,114
Registered: 12/4/04
Re: Basis of module and basis of vector space qs.
Posted: Nov 19, 2013 12:16 AM
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On Monday, November 18, 2013 2:17:05 PM UTC-6, Sandy wrote:
> Is it the case that
>
>
>
> a module has a basis iff it is a free module?


Yes.


> Is it the case that
>
>
>
> (i) every vector space has a basis iff the axiom of choice holds;


Yes.


> (ii) for every vector space V,
>
> (if B and C are bases of V then, there is a bijection: B -> C)
>
> iff the axiom of choice holds


I don't know for sure, but my dim memory is that this is strictly weaker than the full axiom of choice. You might want to check out the website/book on equivalents to the Axiom of Choice by the Rubins.

--
Arturo Magidin



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