Free module

In mathematics, a free module is a free object in a category of modules. Given a set $S$ , a free module on $S$ is a free module with basis $S$ .

Every vector space is free,^[1] and the free vector space on a set is a special case of a free module on a set.

Definition

A free module is a module with a basis:^[2] a linearly independent generating set.

For an $R$ -module $M$ , the set $E\subseteq M$ is a 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 linearly independent, that is, $r_1 e_1 + r_2 e_2 + \cdots + r_n e_n = 0_M$ for $e_1, e_2, \ldots , e_n$ distinct elements of $E$ implies that $r_1 = r_2 = \cdots = r_n = 0_R$ (where $0_M$ is the zero element of $M$ and $0_R$ 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 $x\in M$ .

The definition of an infinite free basis is similar, except that $E$ will have infinitely many elements. However the sum must still 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 $E$ . The module is simply the direct sum of $|E|$ copies of $R$ , often denoted $R^{(E)}$ . We give a concrete realization of this direct sum, denoted by $C(E)$ , as follows:

Carrier: $C(E)$ contains the functions $f:E\to R$ such that $f(x)=0$ for cofinitely many (all but finitely many) $x\in E$ .
Addition: for two elements $f,g\in C(E)$ , we define $f+g\in C(E)$ by $(f+g)(x) = f(x) + g(x), \forall x\in E$ .
Inverse: for $f\in C(E)$ , we define $(-f)\in C(E)$ by $(-f)(x) = -(f(x)), \forall x\in E$ .
Scalar multiplication: for $\alpha\in R, f\in C(E)$ , we define $\alpha f\in C(E)$ by $(\alpha f)(x) = \alpha (f(x)), \forall x\in E$ .

A basis for $C(E)$ is given by the set $\{\delta_a : a\in E\}$ where

\delta_a(x) = \begin{cases} 1, \quad\mbox{if } x=a; \\ 0, \quad\mbox{if } x\neq a \end{cases}

(a variant of the Kronecker delta and a particular case of the indicator function, for the set $\{a\}$ ).

Define the mapping $\iota : E\to C(E)$ by $\iota(a) = \delta_a$ . This mapping gives a bijection between $E$ and the basis vectors $\{\delta_a\}_{a\in E}$ . We can thus identify these sets. Thus $E$ may be considered as a linearly independent basis for $C(E)$ .

Universal property

The mapping $\iota : E\to C(E)$ defined above is universal in the following sense. If there is an arbitrary $R$ -module $M$ and an arbitrary mapping $\varphi : E\to M$ , then there exists a unique module homomorphism $\psi : C(E)\to M$ such that $\varphi = \psi\circ\iota$ .

Generalisations

Many statements about free modules, which are wrong for general modules over rings, are still true for certain generalisations of free modules. Projective modules are direct summands of free modules, so one can choose an injection in a free module and use the basis of this one to prove something for the projective module. Even weaker generalisations are flat modules, which still have the property that tensoring with them preserves exact sequences, and torsion-free modules. If the ring has special properties, this hierarchy may collapse, e.g., for any perfect local Dedekind ring, every torsion-free module is flat, projective and free as well.

See local ring, perfect ring and Dedekind ring.

Notes

↑ Keown (1975). Computational Methods for Modeling of Nonlinear Systems. p. 24.
↑ Hazewinkel (1989). Encyclopaedia of Mathematics, Volume 4. p. 110.

References

Adamson, Iain T. (1972). Elementary Rings and Modules. University Mathematical Texts. Oliver and Boyd. pp. 65–66. ISBN 0-05-002192-3. MR 0345993.
Keown, R. (1975). An Introduction to Group Representation Theory. Mathematics in science and engineering 116. Academic Press. ISBN 978-0-12-404250-6. MR 0387387.
Govorov, V. E. (2001), "Free module", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 .

External links

This article incorporates material from free vector space over a set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.