I am working with the the tensor product $-\otimes_R -$ over some noncommutative ring $R$. Is the tensor product always commutative if $R$ is commutative? If so, can the tensor product be commutative if $R$ is noncommutative?
The tensor product's commutativity depends on the commutativity of the elements. If the ring is commutative, the tensor product is as well. If the ring R is non-commutative, the tensor product will only be commutative over the commutative sub-ring of R. There will always be tensors over the ring that will not commute if R is non-commutative.