Proposition 1.1 Let X be a set, and let R be a unital ring. Then there exists a left R-module F_RX and an injective function i_X: X → FRX such that F_RX is freely generated by i_X(X). The R-module F_RX and the function i_X: X → F_RX then satisfy the following universal property:

given any left R-module M, and given any function f: X → M,  there exists a unique R-module homomorphism ϕ: F_RX → M such that ϕ ◦ i_X = f.

The elements of F_RX may be represented as functions from X to R that have only finitely many non-zero values. Also given any element x of X, the corresponding element i_X(x) of FRX is represented by the function δ_x: X → R, where δ_x maps x to the identity element of R, and maps all other elements of X to the zero element of R.

Proof Let 0_R and 1_R denote the zero element and the multiplicative identity element respectively of the ring R.

We define F_RX to be the set of all functions σ: X → R from X to R that have at most finitely many non-zero values.

Note that if σ and τ are functions from X to R that have at most finitely many non-zero values, then so is the sum σ+τ of the functions σ and τ (where  (σ + τ )(x) = σ(x) + τ (x) for all x ∈ X). Therefore addition of functions is a binary operation on the set F_RX. Moreover F_RX is an Abelian group with respect to the operation of addition of functions.

Given r ∈ R, and given σ ∈ FRX, let rσ be the function from X to R defined such that (rσ)(x) = rσ(x) for all x ∈ X. Then

                        r(σ + τ ) = rσ + rτ, (r + s)σ = rσ + sσ,

                                  (rs)σ = r(sσ), 1_Rσ = σ

for all σ, τ ∈ FRX and r, s ∈ R. It follows that FRX is a module over the ring R.

Given x ∈ X, let δ_x: X → R be the function defined such that

Then δ_x ∈ F_RX for all x ∈ X. We denote by i_X: X → FRX the function that sends x to δ_x for all x ∈ X.

 We claim that F_RX is freely generated by the set i_X(X), where i_X(X) = {δ_x : x ∈ X}. Let M be an R-module, and let f: X → M be a function from X to M. We must prove that there exists a unique R-module homomorphism ϕ: F_RX → M such that ϕ ◦ iX = f (Lemma 1.1)in(Free Modules).

Let σ be an element of F_RX. Then σ is a function from X to R with at most finitely many non-zero values. Then σ =∑_{x∈supp σ}  σ(x)δ_x, where supp σ = {x ∈ X : σ(x) ≠0_R}.

We define ϕ(σ) = ∑_{x∈supp σ} σ(x)f(x). This associates to each element σ of F_RX a corresponding element ϕ(σ) of M. We obtain in this way a function ϕ: F_RX → M.

Let σ and τ be elements of FRX, let r be an element of the ring R, and let Y be a finite subset of X for which supp σ ⊂ Y and supp τ ⊂ Y . Then supp(σ + τ ) ⊂ Y , and

Thus ϕ: FRX → M is the unique R-module homomorphism satisfying ϕ ◦ iX = f.

It now follows from Lemma 8.5 that the R-module FRX is freely generated by i_X(X). We have also shown that the required universal property is satisfied by the module FRX and the function i_X.

Definition Let X be a set, and let R be a unital ring. We define the free left R-module on the set X to be the module FRX constructed as described in the proof of Proposition 1.1. Moreover we may consider the set X to be embedded in the free module F_RX via the injective function iX: X → F_XX described in the statement of that proposition Abelian groups are modules over the ring Z of integers. The construction of free modules therefore associates to any set X a corresponding free Abelian group F_ZX.