Let Q, R and S be unital rings, let M be a Q-R-bimodule, and let N be an R-S-bimodule. Then M is a right R-module and N is a left R-module. We can therefore form the tensor product M ⊗_R N of M and N over the ring R.

This tensor product is an Abelian group under the operation of addition.

           Let q ∈ Q and r ∈ R. The definition of bimodules ensures that (qx)r = q(xr) for all x ∈ M. Let Lq: M ×N → M ⊗R N be the function defined such that L_q(x, y) = (qx) ⊗ y for all x ∈ M and y ∈ N. Then the function f is Z-bilinear. Moreover

                    L_q(xr, y) = (q(xr)) ⊗ y = ((qx)r) ⊗ y = (qx) ⊗ (ry) = Lq(x, ry).

for all x ∈ M and y ∈ N. It follows from Lemma 8.14 that there exists a group homomorphism λ_q: M ⊗_R N → M ⊗_R N, where λ_q(x⊗y) = (qx)⊗y for all x ∈ M and y ∈ N. Similarly, given any element s of the ring S, there exists a group homomorphism ρs: M ⊗_R N → M ⊗_R N, where λ_s(x⊗y) = x⊗(ys).

We define qα = λ_q(α) and αs = ρs(α) for all α ∈ M ⊗_R N. One can check that M ⊗_R N is a Q-S-bimodule with respect to these operations of left multiplication by elements of Q and right multiplication by elements of S. Moreover, given any Q-S-bimodule P, and given any Z-bilinear function f: M × N → P that satisfies

                          f(qx, y) = qf(x, y), f(xr, y) = f(x, ry), f(x, ys) = f(x, y)s

for all x ∈ M, y ∈ N, q ∈ Q, r ∈ R and s ∈ S, there exists a unique Q-S bimodule homomorphism ϕ: M ⊗R N → P such that f(x, y) = ϕ(x ⊗ y) for all x ∈ M and y ∈ N.