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Free module

April 16, 2012

Free modules are another instance of free objects. (see this post for a discussion on free objects and the construction of the free group.) They are the modules satisfying a universal property analogue to that of free groups. Here is a more constructive definition:

Definition: Let S be a set and R be a ring. An R-module F is free on S if for all 0 \neq x \in F, there exist unique non-zero r_1, \ldots r_n \in R and unique non-zero s_1, \ldots, s_n \in S such that x = \sum_{i=1}^nr_is_i for some n \in \textbf{N}. The set S is said to be a basis or a set of free generators for F. If R is commutative, we call |S| the rank of F.

Note that there is a natural injection \iota : S \to F identifying s \in S with 1_Rs \in F.

This definition coincides with our intuition that the only relations among elements of F should be the minimal relations required for F to be a module. For example, the integers modulo n, \mathbf{Z}_n is not a free \mathbf{Z}-module on any set. Indeed, although any element x\in \mathbf{Z}_n can be uniquely written as an element of the underlying set, we have x = (n+1)x = (2n+1)x = \cdots so we don’t have unicity in the ring elements, as the definition requires.

To construct a free R-module on a set S, consider the free abelian group on S, which you can obtain by quotienting the free group on S, FG(S) by the normal subgroup generated by the set of all elements of the form ab - ba in FG(S). (This will satisfy the analog universal property for abelian groups.) This group can be identified with

\textbf{Ab}(S) := \{ \displaystyle\sum_{finite} \epsilon_is_i \mid s_i \in S; \epsilon_i \in \{ -1_R, 1_R \} \}

where the operation is still concatenation of words but written additively. We now naturally define the free R-module over A by

F(S) = \{ \displaystyle\sum_{finite}r_is_i \mid r_i \in R; s_i \in S \}

with the R-action defined as r(\sum r_is_i) = \sum (rr_i)s_i. This is easily seen to satisfy the defining properties of a module and the universal property of free modules:

Theorem: (Universal property) For S a set and R a unitary ring, the R-module F(S) constructed above satisfy the universal property of free modules. In other words, for every R-module M and every map \varphi : S \to M, there exists a unique R-module homomorphism \phi : F(S) \to M such that the following diagram commutes:

Proof: Let \iota : S \hookrightarrow F(S) be the natural inclusion \iota(s) = 1s. Define \phi : F(S) \to M on \iota(S) \subset F(S) as \phi(1s) = \varphi(s). Then, since \iota(S) generates F(S), \phi extends uniquely by linearity to an R-module homomorphism on F(S) and we have \varphi = \phi \circ \iota as wanted.

QED

Note that as in the case of the free groups, this property uniquely characterizes free modules up to isomorphism.

Reference: D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed.

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