Let R be a unital ring that is not necessarily commutative, let M be a right R-module, and let N be a left R-module. These modules are Abelian groups under the operation of addition, and Abelian groups are modules over the ring Z of integers. We can therefore form their tensor product M ⊗_Z N. This tensor product is an Abelian group.

Let K be the subgroup of M ⊗Z N generated by the elements

                 (xr) ⊗_Z y − x ⊗_Z (ry)

for all x ∈ M, y ∈ N and r ∈ R, where x ⊗_Z y denotes the tensor product of x and y in the ring M ⊗_Z N. We define the tensor product M ⊗_R N of the right R-module M and the left R-module N over the ring R to be the quotient group M ⊗_Z N/K. Given x ∈ M and y ∈ N, let x ⊗ y denote the

image of x ⊗_Z y under the quotient homomorphism π: M ⊗_Z N → M ⊗_R N.

Then

                (x₁ + x₂) ⊗ y = x₁ ⊗ y + x₂ ⊗ y, x ⊗ (y₁ + y₂) = x ⊗ y₁ + x ⊗ y₂,

and

                                   (xr) ⊗ y = x ⊗ (ry)

for all x, x₁, x₂ ∈ M, y, y₁, y₂ ∈ N and r ∈ R.

Lemma 1.1 Let R be a unital ring, let M be a right R-module, and let N be a left R-module. Then the tensor product M ⊗_R N of M and N is an Abelian group that satisfies the following universal property:

given any Abelian group P, and given any Z-bilinear function

f: M × N → P which satisfies

                     f(xr, y) = f(x, ry)