Definition Let F be a left module over a unital ring R, and let X be a subset of F. We say that the left R-module F is freely generated by the subset X if, given any left R-module M, and given any function f: X → M,  there exists a unique R-module homomorphism ϕ: F → M that extends the function f.

Example Let K be a field. Then a K-module is a vector space over K. Let V be a finite-dimensional vector space over the field K, and let b₁, b₂, . . . , b_n be a basis of V . Then V is freely generated (as a K-module) by the set B,  where B = {b₁, b₂, . . . , b_n}. Indeed, given any vector space W over K, and given any function f: B → W, there is a unique linear transformation ϕ: V → W that extends f. Indeed

for all λ₁, λ₂, . . . , λ_n ∈ K. (Note that a function between vector spaces over some field K is a K-module homomorphism if and only if it is a linear transformation.)

Definition A left module F over a unital ring R is said to be free if there exists some subset of F that freely generates the R-module F.

Lemma 1.1 Let F be a left module over a unital ring R, let X be a set, and let i: X → F be a function. Suppose that the function f: X → F satisfies 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 → M such that ϕ ◦ i = f.

Then the function i: X → F is injective, and F is freely generated by i(X).

Proof Let x and y be distinct elements of the set X, and let f be a function satisfying f(x) = 0_R and f(y) = 1_R, where 0_R and 1_R denote the zero element and the multiplicative identity element respectively of the ring R.

The ring R may be regarded as a left R-module over itself. It follows from the universal property of i: X → M stated above that there exists a unique R-module homomorphism θ: F → R for which θ ◦ i = f. Then θ(i(x)) = 0_R and θ(i(y)) = 1_R. It follows that i(x) ≠i(y). Thus the function i: X → F is injective.

Let M be a left R-module, and let g:i(X) → M be a function defined on i(X). Then there exists a unique homomorphism ϕ: F → M such that ϕ ◦ i = g ◦ i. But then ϕ|i(X) = g. Thus the function g:i(X) → M extends uniquely to a homomorphism ϕ: F → M. This shows that F is freely generated by i(X), as required.

Let F₁ and F₂ be left modules over a unital ring R, let X₁ be a subset of F₁, and let X₂ be a subset of F₂. Suppose that F1 is freely generated by X₁,  and that F₂ is freely generated by X₂. Then any function f: X₁ → X₂ from X₁ to X₂ extends uniquely to a R-module homomorphism from F₁ to F₂. We denote by f]: F₁ → F₂ the unique R-module homomorphism that extends f.

Now let F₁, F₂ and F₃ be left modules over a unital ring R, and let X₁,  X₂ and X₃ be subsets of F₁, F₂ and F₃ respectively. Suppose that the left R-module F_i is freely generated by X_i for i = 1, 2, 3. Let f: X₁ → X₂ andg: X₂ → X₃ be functions. Then the functions f, g and g ◦ f extend uniquely to R-module homomorphisms f] : F₁ → F₂, g] : F₂ → F₃ and (g ◦f)]: F₃ → F₃.