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Math Help - Modules

  1. #1
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    Modules

    Given a set X, prove that there exists a free R-module F with a basis B for which there is a
    bijection ϕ : B → X.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Modules

    Quote Originally Posted by jcir2826 View Post
    Given a set X, prove that there exists a free R-module F with a basis B for which there is a
    bijection ϕ : B → X.
    Consider the free module R^{\oplus X} (i.e. X-fold coproduct). This has a natural basis of the form \{e_x:x\in X\} where e_x is the tuple with 1 in the x^{\text{th}} coordinate and zero elsewhere. I think the rest should be obvious.
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  3. #3
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    Re: Modules

    So I am just missing to show that there is a bijection from this natural basis to the free module direct summand R and X?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Re: Modules

    Quote Originally Posted by jcir2826 View Post
    So I am just missing to show that there is a bijection from this natural basis to the free module direct summand R and X?
    The basis is \{e_x:x\in X\} and isn't x\mapsto e_x a bijection?
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