Free module
|
|
In mathematics, a free module is a module having a free basis.
For an R-module M, the set E = {e1, e2, ... en} is a free basis for M if and only if:
- 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;
- 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 M has a free basis with n elements, then M is said to be free of rank n, or more generally free of finite rank.
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 E → R such that the preimage of R \ {0} is finite.
- 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∈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.