Lemma 1.1 Let L, M and N be R-modules over a unital commutative ring R. Then

                            (L ⊕ M) ⊗_R N ≅ (L ⊗_R N) ⊕ (M ⊗_R N).

Proof The function

                       j: (L ⊕ M) × N → (L ⊗_R N) ⊕ (M ⊗_R N)

is an R-bilinear function, where j((x, y), z) = (x ⊗ z, y ⊗ z) for all x ∈ L,  y ∈ M and z ∈ N. We prove that the R-module (L ⊗R N) ⊕ (M ⊗R N) and the R-bilinear function j satisfy the universal property that characterizes the tensor product of (L ⊕ M) and N over the ring R up to isomorphism.

Let P be an R-module, and let f: (L ⊕ M) × N → P be an R-bilinear function. Then f determines R-bilinear functions g: L × N → P and h: M × N → P, where g(x, z) = f((x, 0), z) and h(y, z) = f((0, x), z for all x ∈ L,  y ∈ M and z ∈ N. Moreover

                      f((x, y), z) = f((x, 0)+(0, y), z) = f((x, 0), z)+f(0, y), z) = g(x, z)+h(y, z).

for all x ∈ L, y ∈ M and z ∈ N. Now there exist unique R-module homomorphisms ϕ: L ⊗_R N → P ψ: L ⊗_R N → P satisfying the identities ϕ(x ⊗ z) = g(x, z) and ψ(y ⊗ z) = h(y, z) for all x ∈ L, y ∈ M and z ∈ N.

Then

                   f((x, y), z) = ϕ(x ⊗ z) + ψ(y ⊗ z) = θ((x ⊗ z),(y ⊗ z)) = θ(j((x, y), z),

where θ: (L⊗_R N)⊕(M ⊗_R N) → P is the R-module homomorphism defined such that θ(u, v) = ϕ(u) + ψ(v) for all u ∈ L ⊗_R N and v ∈ M ⊗_R N.

We have thus shown that, given any R-module P, and given any R-bilinear function f: (L ⊕ M) × N → P, there exists an R-module homomorphism θ: (L ⊗_R N) ⊕ (M ⊗_R N) → P satisfying f = theta ◦ j. This homomorphism is uniquely determined. It follows directly from this that

                              (L ⊕ M) ⊗_R N ≅ (L ⊗_R N) ⊕ (M ⊗_R N),