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Math Help - Tensor Products of R-modules

  1. #1
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    Tensor Products of R-modules

    If I'm working with R-module tensors, is it true that

    (s1 x r1)(s2 x r2)= (s1s2 x r1r2), (where x should have a circle around it)?

    The question comes from this:
    Establish an isomorphism between two R-algebras « Project Crazy Project .

    Also, why are we sure 1 is in R? (It looks like they're assuming this is true, b/c to my knowledge R[x] is the set of polynomials with coefficients from R, but they have (1x1) at one point, and (s x x^i). )
    Last edited by gummy_ratz; November 21st 2011 at 03:33 PM.
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    MHF Contributor Drexel28's Avatar
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    Re: Tensor Products of R-modules

    Quote Originally Posted by gummy_ratz View Post
    If I'm working with R-module tensors, is it true that

    (s1 x r1)(s2 x r2)= (s1s2 x r1r2), (where x should have a circle around it)?

    The question comes from this:
    Establish an isomorphism between two R-algebras « Project Crazy Project .

    Also, why are we sure 1 is in R? (It looks like they're assuming this is true, b/c to my knowledge R[x] is the set of polynomials with coefficients from R, but they have (1x1) at one point, and (s x x^i). )
    This is how we naturally define the ring operation on the tensor product of algebras, so yes. And yes, R[x] is unital, and we are assuming that that R is a unital subring.
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