Distributivity of tensor product over direct sum

In this post, I will show that the direct sum and tensor product of modules fit nicely together. Recall that for an -module, is the (internal) direct sum of two of its submodules if and only if every element can be uniquely written as with . We can naturally identify the direct sum of two -modules with

Theorem: Let be -modules for a commutative (and unitary) ring. Then there are unique isomorphisms

Proof: I will only give the first isomorphism, the other one being analogous. Consider the map from to which takes to This map is easily seen to be bilinear and so by the universal property of tensor products, it induces a unique -module homomorphism  such that

In the other direction, both maps, respectively from and to respectively defined as and are bilinear. Thus, they induce unique -module homomorphisms

such that and Therefore, defined as

is a well-defined -module homomorphism.

It now suffices to verify that and are inverses and we are done.

QED

Note that this result extends easily by induction to any finite direct sums. More generally, it can be shown that it is also true for arbitrary direct sums so that we have

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

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