In mathematics, a free module is a free object in the category of modules. Given a set S, a free module on S is a (particular construction of a) free module with basis S. Every vector space is free, and the free vector space on a set is a special case of a free module on a set.
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Definition
A free module is a module with a free basis: a linearly independent generating set.
For an R-module M, the set E = {e1, e2, ... en} is a free basis for M if:
- E is a generating set for M, that is to say every element of M is a finite sum of elements of E multiplied by coefficients in R;
- E is a free set that is if r1e1 + r2e2 + ... + rnen = 0, then r1 = r2 = ... = rn = 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 cofinitely many (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 { Δa : a ∈ E } where
- <math> \Delta_a(x) = \begin{cases} 1, \quad\mbox{if } x=a; \\ 0, \quad\mbox{if } x\neq a. \end{cases} </math>
Define the mapping ι : E → C(E) by ι(a) = Δa. This mapping gives a bijection between E and the basis vectors {Δa}a∈E. 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
This article incorporates material from free vector space over a set on PlanetMath, which is licensed under the GFDL.
