Let be a commutative ring with 1, and let be an -bimodule in the usual way. (I.e. .) Prove that if is cyclic as an -module, then the tensor algebra is commutative.
Note that as a ring, is generated by 0- and 1-tensors. Thus to show that is commutative, it suffices to show that these generators commute pairwise.
Certainly the 0-tensors commute with each other, since is commutative. Similarly, 0- and 1-tensors commute pairwise using the “commutativity condition” . It remains to be seen that 1-tensors commute with each other. To that end, suppose , and let . Note that , and indeed the 1-tensors commute. Thus is a commutative -algebra.
In particular, we have for all and , so that .