Associativity and k-fold tensor products

Here I will use the universal property of tensor products regarding bilinear maps to prove that the construction of tensor products is associative. For simplicity, I will take the ring to be commutative so that we can view -modules as -bimodules.

Theorem: Let be a commutative and unitary ring, and be -modules. Then there is a unique isomorphism

Proof: For a fixed it is easy to see that the map such that is bilinear. For example we have By the universal property of tensor products, for every such fixed we have a unique -module homomorphism such that Thus, the map such that (and then extended by linearity on the first argument) is well-defined. Moreover, is easily seen to be bilinear and thus (again by the universal property) induces a unique -module homomorphism taking to Similarly, we can construct a homomorphism the other way around, inverse to this one, hence the isomorphism.

QED

We thus see that for a commutative ring we can unambiguously construct the k-fold tensor product of k -modules Moreover, from the universal property of tensor product and the last remark in the last post, we deduce that this k-fold tensor product is the universal object with respect to k-multilinear maps :

Theorem (Universal property): Let be a commutative ring and let be -modules. Let be such that Let be a k-multilinear function. Then there exists a unique such that

Once again, we deduce the existence of a bijection between the set of k-multilinear functions from to and the set of -module homomorphisms from to

Reference: D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed.

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