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Math Help - Tensor Product of Modules

  1. #1
    Senior Member slevvio's Avatar
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    Tensor Product of Modules

    Hello all, I was wondering if I could get some help proving the following statement:

    \mathbb{Z} / 8 \otimes_{\mathbb{Z}} \mathbb{Z}_{\langle 2 \rangle} = \mathbb{Z} / 8

    where \mathbb{Z}_{\langle 2 \rangle} is the integers localised at the prime ideal <2>.

    Can anyone give me some hints as to how to calculate this?
    Any help would be appreciated.
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  2. #2
    MHF Contributor

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    Re: Tensor Product of Modules

    Quote Originally Posted by slevvio View Post
    Hello all, I was wondering if I could get some help proving the following statement:

    \mathbb{Z} / 8 \otimes_{\mathbb{Z}} \mathbb{Z}_{\langle 2 \rangle} = \mathbb{Z} / 8

    where \mathbb{Z}_{\langle 2 \rangle} is the integers localised at the prime ideal <2>.

    Can anyone give me some hints as to how to calculate this?
    Any help would be appreciated.
    let's consider a more general case: let R be a commutative ring with 1 and let I be an ideal of R. let S be a multiplicatively closed subset of R with 1 \in S and 0 \notin S. suppose that every element of S is invertible modulo I, i.e. s + I is an invertible element of R/I for all s \in S. then

    S^{-1}(R/I) \cong R/I,

    because the natural R-module homomorphism f : R/I \longrightarrow S^{-1}(R/I) defined by f(x)=x/1 is an isomorphism (why?). thus

    (R/I) \otimes_R S^{-1}R \cong S^{-1}(R/I) \cong R/I.
    Last edited by NonCommAlg; December 9th 2011 at 12:44 PM.
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