Let R and S be unital rings with multiplicative identity elements 1_R and 1_S,  and let S op be the unital ring (S, +, ×^-) whose elements are those of S, whose operation of addition is the same as that defined on S, and whose operation×^- of multiplication is defined such that s₁×^-s₂ = s₂s₁ for all s₁, s₂ ∈ S.

We can then construct a ring R ⊗_Z S^op. The elements of this ring belong to the tensor product of the rings R and S^op over the ring Z of integers, and the operation of addition on R ⊗_Z S op is that defined on the tensor product.

The operation of multiplication on R ⊗_Z S^op is then defined such that

                   (r₁ ⊗ s₁) × (r₂ ⊗ s₂) = (r₁r₂) ⊗ (s₁×^-s₂) = (r₁r₂) ⊗ (s₂s₁).

Lemma 1.1  Let R and S be unital rings, and let M be an R-S-bimodule.

Then M is a left module over the ring R ⊗_ZS^op, where

                                (r₁ ⊗ s₁) × (r₂ ⊗ s₂) = (r₁r₂) ⊗ (s₂s₁)

for all r₁, r₂ ∈ R and s₁, s₂ ∈ S, and where

                                                  (r ⊗ s).x = (rx)s = r(xs)

for all r ∈ R, s ∈ S and x ∈ M.

Proof Given any element x of M, let b_x: R × S → M be the function defined such that b_x(r, s) = (rx)s = r(xs) for all r ∈ R and s ∈ S. Then the function b_x is Z-bilinear, and therefore induces a unique Z-module homomorphismβ_x: R ⊗_Z S^op → M, where β_x(r ⊗ s) = b_x(r, s) = (rx)s for all r ∈ R,  s ∈ S and x ∈ M. We define u.x = β_x(u) for all u ∈ R ⊗_Z S^op and x ∈ M.

Then (u₁ + u₂).x = u₁.x + u₂.x for all u₁, u₂ ∈ R ⊗_Z S^op and x ∈ M, because β_x is a homomorphism of Abelian groups. Also u.(x₁ + x₂) = u.x₁ + u.x₂,  because b_x1+x2 = b_x1 + b_x2 and therefore β_x1+x2 = β_x1 + β_x2.

Now

                    (r₁ ⊗ s₁).((r₂ ⊗ s₂).x) = (r₁ ⊗ s₁).((r₂x)s₂) = r₁(r₂(xs₂))s₁

                                                               = ((r₁r₂)(xs₂))s₁ = (r₁r₂)((xs₂)s₁)

                                                              = (r₁r₂)(x(s₂s₁) = ((r₁r₂) ⊗_Z (s₂s₁)).x

                                                             = ((r₁ ⊗_Z s₁) × (r₂ ⊗_Z s₂)).x

for all r₁, r₂ ∈ R, s₁, s₂ ∈ S and x ∈ M. The bilinearity of the function β_x then ensures that u₁.(u₂.x) = (u₁×u₂).x for all u₁, u₂ ∈ R⊗_Z S^op and x ∈ M.

Also (1_R, 1_S).x = x for all x ∈ M, where 1_R and 1_S denote the identity elements of the rings R and S. We conclude that M is a left R ⊗_Z S^op, as required.

Let R and S be unital rings, and let M be a left module over the ring R ⊗_Z S^op. Then M can be regarded as an R-S-bimodule, where (rx)s = r(xs) = (r ⊗ s).x for all r ∈ R, s ∈ S and x ∈ M. We conclude therefore that all R-S-bimodules are left modules over the ring R ⊗_Z S op, and vica versa. It follows that any general result concerning left modules over unital rings yields a corresponding result concerning bimodules.