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[Module theory] Prove that something forms a left R module.

  1. Apr 23, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose that R and S are two rings, M, is a (R-S) bi-module and N is a left R-module. Show that [itex] M \otimes N [/itex] has the structure of a left S-module.

    3. The attempt at a solution

    Well, [itex] M\otimes N [/itex] is an Abelian group, so it's enough that I define a scalar product on [itex] M\otimes N [/itex]. I'm thinking of defining:
    [itex] s.\sum_{i=1}^t{x_i\otimes y_i} = \sum_{i=1}^t{sx_i\otimes y_i} [/itex]
    Now I'm a bit clueless about how I should show that this scalar multiplication is well-defined. I know that I should suppose [itex] \sum_{i=1}^t{x_i\otimes y_i} = \sum_{i=1}^{t'}{x'_i\otimes y'_i} [/itex] and then show that [itex] \sum_{i=1}^t{sx_i\otimes y_i} = \sum_{i=1}^{t'}{sx'_i\otimes y'_i} [/itex] but I don't know how I should do that. I'm looking for a nice map from [itex]F[/itex] to [itex]F/K = M\otimes N[/itex] that does the trick but nothing good comes to my mind now. Any ideas would be appreciated greatly.
    Thanks in advance.
     
  2. jcsd
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