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Date: 25-7-2021
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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),
as required.
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