Let be a commutative ring with 1 and let be a unital -bimodule such that . Prove that if is an -algebra such that for all and if is an -module homomorphism, then there exists a unique -algebra homomorphism such that .
By the universal property of tensor algebras, there exists a unique -module homomorphism such that . Now recall that is the ideal of generated by simple tensors of the form . Note that in , so that . By the first isomorphism theorem for -algebras, we have a well-defined -algebra homomorphism given by . Certainly then .
To see uniqueness, suppose is another such homomorphism. Then .