The tensor product between modules A and B is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring R. The familiar formulas hold, but now α is any element of R, (a_1 + a_2)⊗b = a_1 ⊗b + a_2 ⊗b a⊗(b_1 + b_2) = a⊗b_1 + a⊗b_2 α(a⊗b) = (α a)⊗b = a⊗(α b). This generalizes the definition of a tensor product for vector spaces since a vector space is a module over the scalar field. Also, vector bundles can be considered as projective modules over the ring of functions, and group representations of a group G can be thought of as modules over C G. The generalization covers those kinds of tensor products as well.
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