Tensor Product of Bimodules

Let $R,S,T$ be Rings, $(M,\lambda_{M},\rho_{M})$ a R-S-bimodule, and $(N,\lambda_{N},\rho_{N})$ a S-T-bimodule. The tensor product $M\otimes_{S}N$ is an R-T-bimodule defined by quotienting the Tensor Product of Abelian Groups by the congruence generated from:

• $\rho(m,s)\otimes n\sim m\otimes\lambda(s,n)$

where $m\otimes n$ denotes a generator of the aforementioned Tensor Product of Abelian Groups. Note that this is essentially the Composition of Profunctors.

If we unfold all this abstraction, we can see that elements of the tensor product are essentially finite linear combinations of "pure tensors" $m\otimes s\otimes n$, though many authors will omit the $s$ and simply write $m\otimes n$.

Universal Property

If we examine the quotient from before, we can note that we are forcing the tensor map $\iota:M\times N\to M\otimes_{S}N$ to be Balanced. This yields the following universal property:

• Every Balanced map $M\times N\to H$ factors uniquely as
  with $\overline{\varphi}$ a Linear Map.

Examples

• Let $G$ be an Abelian Group, regarded as a $(\mathbb{Z},\mathbb{Z})$-Bimodule. Moreover, consider the Cyclic Group $\mathbb{Z}/n\mathbb{Z}$; this is a Left Module over itself, so we can then form the tensor product $\mathbb{Z}/n\mathbb{Z}\otimes G$, which is sometimes called "the reduction of $G$ mod $n$". This is isomorphic to $G/nG$, which kills off all $n$-degree Torsion.