Linear Algebra over a Ring/Multilinear algebra

Definition (multilinear function):

Let be a ring, and let be -modules. Then the set of -multilinear functions from to is the set

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Proposition (equivalent definition of tensor product of free modules using multilinear functions):

Let be a ring, and let be free, finitely generated -modules. Then if we alternatively define


and let the elementary tensors be , then the from this definition satisfies the same universal property as the usual tensor product . In particular, the two tensor products are canonically isomorphic.

Proof: For , let be a basis of , where is the respective finite index set. Given any -module and any multilinear map , we want a unique linear function such that , where is the map that sends a tuple to the respective elementary tensor.