In mathematics, a free module is a module having a free basis. Free modules are examples of free objects.

For an R-module M, the set E = {e₁, e₂, ... e_n} is a free basis for M if:

1. E is a generating set for M, that is to say every element of M is a sum of elements of E multiplied by coefficients in R;
2. E is a free set that is if r₁e₁ + r₂e₂ + ... + r_ne_n = 0, then r₁ = r₂ = ... = r_n = 0 (where 0 is the zero element of M and 0 is the zero element of R).

If R has invariant basis number, then by definition any two bases have the same cardinality. The cardinality of any (and therefore every) basis is called the rank of the free module M, and M is said to be free of rank n, or simply free of finite rank if the cardinality is finite.

Note that an immediate corollary of (2) is that the coefficients in (1) are unique for each x.

The definition of an infinite free basis is similar, except that E will have infinitely many elements. However the sum must be finite, and thus for any particular x only finitely many of the elements of E are involved.

In the case of an infinite basis, the rank of M is the cardinality of E.

Construction

Given a set E, we can construct a free R-module over E, denoted by C(E), as follows:

• As a set, C(E) contains the functions f : E → R such that f(x) = 0 for all but finitely many x in E.
• Addition: for two elements f, g ∈ C(E), we define f + g ∈ C(E) by (f + g)(x) = f(x) + g(x) for all x ∈ E.
• Scalar multiplication: for α ∈ R and f ∈ C(E), we define αf ∈ C(E) by (αf)(x) = αf(x) for all x ∈ E.

A basis for C(E) is given by the set { Δ_a : a ∈ E } where

$\backslash Delta\_a(x)\; =\; \backslash begin\{cases\}\; 1,\; \backslash quad\backslash mbox\{if\; \}\; x=a;\; \backslash \backslash \; 0,\; \backslash quad\backslash mbox\{if\; \}\; x\backslash neq\; a.\; \backslash end\{cases\}$

Define the mapping ι : E → C(E) by ι(a) = Δ_a. This mapping gives a bijection between E and the basis vectors {Δ_a}_a∈X. We can thus identify these spaces. Then E becomes a linearly independent basis for C(E).

Universal property

The mapping ι : E → C(E) defined above is universal in the following sense. If φ is an arbitrary mapping from E to some R-module M, then there exists a unique mapping ψ: C(E) → M such that φ = ψ o ι.

See also

• Free object

Module theory | Free algebraic structures

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