Let be a ring, let be a nonempty set, let be a right -module, and let be a family of left -modules. Prove that as abelian groups, .
We will make use of the universal properties of tensor products and direct sums to prove this without too many fine details.
Recall that for each , we have a canonical injection . Thus for all we have a group homomorphism . By the universal property of direct sums, we have a (unique) group homomorphism such that .
Now define by . This mapping is well defined because for a fixed , finitely many are nonzero. Evidenly is bilinear, and so induces a group homomorphism .
We claim that and are mutual inverses. To see this, note that and .
Thus both and are group isomorphisms, and we have .