Definition Let R be a unital commutative ring, and let let M and N and P be R-modules. A function f: M × N → P is said to be R-bilinear if

                                      f(x₁ + x₂, y) = f(x₁, y) + f(x₂, y),

                                      f(x, y₁ + y₂) = f(x, y₁) + f(x, y₂),

and

                                     f(rx, y) = f(x, ry) = rf(x, y)

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

Proposition 1.1 Let R be a unital commutative ring, and let M and N be modules over R. Then there exists an R-module M ⊗R N and an R-bilinear function jM×N : M × N → M ⊗R N, where M ⊗R N and jM×N satisfy the following universal property:

              given any R-module P, and given any R-bilinear function f: M × N → P, there exists a unique R-module homomorphism θ: M ⊗R N → P such that f = θ ◦ j_M×N .

Proof Let FR(M × N) be the free R-module on the set M × N, and let i_M×N : M ×N → F_R(M ×N) be the natural embedding of M ×N in F_R(M ×N). Then, given any R-module P, and given any function f: M × N → P,  there exists a unique R-module homomorphism ϕ: F_R(M × N) → P such that ϕ ◦ i_M×N = f (Proposition 1.1)in(Construction of Free Modules).

Let K be the submodule of F_R(M × N) generated by the elements

                        i_M×N (x₁ + x₂, y) − i_M×N (x₁, y) − i_M×N (x₂, y),

                        i_M×N (x, y₁ + y₂) − i_M×N (x, y₁) − i_M×N (x, y₂),

                                   i_M×N (rx, y) − ri_M×N (x, y),

                                   i_M×N (x, ry) − ri_M×N (x, y)

for all x, x₁, x₂ ∈ M, y, y₁, y₂ ∈ N and r ∈ R. Also let M ⊗R N be the quotient module _FR(M×N)/K, let π: F_R(M×N) → M⊗R_N be the quotient homomorphism, and let j_M×N : M×N → M⊗_RN be the composition function π ◦ i_M×N . Then

                   j_M×N (x₁ + x₂, y) − j_M×N (x₁, y) − j_M×N (x₂, y)

                         = π(i_M×N (x₁ + x₂, y) − i_M×N (x₁, y) − i_M×N (x₂, y)) = 0

for all x₁, x₂ ∈ M and y ∈ N. Similarly

                   j_M×N (x, y₁ + y₂, y) − j_M×N (x, y₁) − j_M×N (x, y₂) = 0

for all x ∈ M and y₁, y₂ ∈ N, and

               j_M×N (rx, y) − rj_M×N (x, y) = π(i_M×N (rx, y) − ri_M×N (x, y)) = 0,

              j_M×N (x, ry) − rj_M×N (x, y) = π(i_M×N (x, ry) − ri_M×N (x, y)) = 0

for all x ∈ M, y ∈ N and r ∈ R. It follows that

                           j_M×N (x₁ + x₂, y) = j_M×N (x₁, y) + j_M×N (x₂, y),

                       j_M×N (x, y₁ + y₂) = j_M×N (x, y₁) + j_M×N (x, y₂),

and

                        j_M×N (rx, y) = j_M×N (x, r_y) = rj_M×N (x, y)

for all x, x₁, x₂ ∈ M, y, y₁, y₂ ∈ N and r ∈ R. Thus j_M×N : M ×N → M ⊗_R N is an R-bilinear function.

Now let P be an R-module, and let f: M × N → P be an R-bilinear function. Then there is a unique R-module homomorphism ϕ: F_R(M ×N) →P such that f = ϕ ◦ i_M×N . Then

                 ϕ(i_M×N (x₁ + x₂, y) − i_M×N (x₁, y) − i_M×N (x₂, y))

                             = f(x₁ + x₂, y) − f(x₁, y) − f(x₂, y) = 0

for all x₁, x₂ ∈ M and y ∈ N. Similarly

                           ϕ(i_M×N (x, y₁ + y₂) − i_M×N (x, y₁) − i_M×N (x, y₂)) = 0

for all x ∈ M and y₁, y₂ ∈ N, and

                  ϕ(i_M×N (rx, y) − ri_M×N (x, y)) = f(rx, y) − rf(x, y) = 0,

                  ϕ(i_M×N (x, ry) − ri_M×N (x, y)) = f(x, ry) − rf(x, y) = 0

for all x ∈ M, y ∈ N and r ∈ R. Thus the submodule K of F_R(M × N)  is generated by elements of ker ϕ, and therefore K ⊂ ker ϕ. It follows that ϕ: F_R(M×N) → P induces a unique R-module homomorphism θ: M⊗R_N → P, where M ⊗R _N = F_R(M × N)/K, such that ϕ = θ ◦ π. Then

                      θ ◦ j_M×N = θ ◦ π ◦ i_M×N = ϕ ◦ i_M×N = f.

Moreover is ψ: M ⊗_R N → P is any R-module homomorphism satisfying ψ ◦ j_M×N = f then ψ ◦ π ◦ i_M×N = f. The uniqueness of the homomorphism ϕ: F_R(M × N) → P then ensures that ψ ◦ π = ϕ = θ ◦ π. But then ψ = θ,  because the quotient homomorphism π: F_R(M ×N) → M ⊗R _N is surjective.                     Thus the homomorphism θ is uniquely determined, as required.

        Let M and N be modules over a unital commutative ring R. The module M ⊗_R N constructed as described in the proof of Proposition 1.1 is referred to as the tensor product M ⊗R N of the modules M and N over the ring R.

Given x ∈ M and y ∈ N, we denote by x⊗y the image j(x, y) of (x, y) under the bilinear function jM×N : M × N → M ⊗_R N. We call this element thetensor product of the elements x and y. Then

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

and

                             (rx) ⊗ y = x ⊗ (ry) = r(x ⊗ y)

for all x, x₁, x₂ ∈ M, y, y₁, y₂ ∈ N and r ∈ R. The universal property characterizing tensor products described in Proposition 1.1 then yields the following result.

Corollary 1.2 Let M and N be modules over a unital commutative ring R,  let M ⊗R N be the tensor product of M and N over R. Then, given any Rmodule P, and given any R-bilinear function f: M × N → P, there exists a unique R-module homomorphism θ: M⊗_RN → P such that θ(x⊗y) = f(x, y)  for all x ∈ M and y ∈ N.

The following corollary shows that the universal property stated in Proposition 1.1 characterizes tensor products up to isomorphism.

Corollary 1.3 Let M, N and T be modules over a unital commutative ring R, let M ⊗R N be the tensor product of M and N, and let k: M ×N → T be an R-bilinear function. Suppose that k: M ×N → T satisfies the universal property characterizing tensor products so that, given any R-module P, and given any R-bilinear function f: M ×N → P, there exists a unique R-module homomorphism ψ: T → P such that f = ψ◦k. Then T ≅ M ⊗_R N, and there is a unique R-isomorphism ϕ: M ⊗_R N → T such that k(x, y) = ϕ(x ⊗_R y)  for all x ∈ M and y ∈ N.

Proof It follows from Corollary 8.8 that there exists a unique R-module homomorphism ϕ: M ⊗_R N → T such that k(x, y) = ϕ(x ⊗ y) for all x ∈ M and y ∈ N. Also universal property satisfied by the bilinear function k: M × N → T ensures that there exists a unique R-module homomorphism ψ: T → M ⊗_R N such that x ⊗ y = ψ(k(x, y)) for all x ∈ M and y ∈ N.

Then ψ(ϕ(x ⊗ y)) = x ⊗ y for all x ∈ M and y ∈ M. But the universal property characterizing the tensor product ensures that any homomorphism from M ×_R N to itself is determined uniquely by its action on elements of the form x ⊗ y, where x ∈ M and y ∈ N. It follows that ψ ◦ ϕ is the identity automorphism of M ⊗R N. Similarly ϕ◦ψ is the identity automorphism of T.

It follows that ϕ: M⊗_RN → T is an isomorphism of R-modules whose inverse is ψ: T → M ⊗_R N. The isomorphism ϕ has the required properties.

Corollary 1.4 Let M be a module over a unital commutative ring R, and let κ: R ⊗_R M → M be the R-module homomorphism defined such that κ(r ⊗ x) = rx for all r ∈ R and x ∈ M. Then κ is an isomorphism, and thus R ⊗_R M ≅ M.

Proof Let P be an R-module, and let f: R × M → P be an R-bilinear function. Let ψ: M → P be defined such that ψ(x) = f(1_R, x) for all x ∈ M, where 1_R denotes the identity element of the ring R. Then ψ is an R-module homomorphism. Moreover f(r, x) = rf(1_R, x) = f(1_R, rx) = ψ(rx) for all x ∈ M and r ∈ R. Thus f = ψ ◦ k, where k: R × M → M is the R-bilinear function defined such that k(r, x) = rx for all r ∈ R and x ∈ M. The result therefore follows on applying Corollary 1.3.

Corollary 1.5 Let M, M`, N and N` be modules over a unital commutative ring R, and let ϕ: M → M`and ψ: N → N` be R-module homomorphisms.

Then ϕ and ψ induce an R-module homomorphism ϕ⊗ψ: M ⊗_R N → M` ⊗<sub>R</sub>N` , where (ϕ ⊗ ψ)(m ⊗ n) = ϕ(m) ⊗ ψ(n) for all m ∈ M and n ∈ N.

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