Show that and are isomorphic as left -modules.
Let be a simple tensor in . We have . In particular, every simple tensor (hence every element of ) can be written as where . This gives (by the universal property of free modules) a unique -module homomorphism such that . Certainly is surjective.
Certainly every simple tensor (and every element) in can be written in the form . By the universal property of free modules, this yields a surjective -module homomorphism .
Now define by . Certainly is -bilinear (and also -bilinear), and so induces a -module homomorphism such that . Certainly is surjective, and so and are isomorphisms. Thus .