Free module

(Redirected from Rank of a free module)

In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module,[1] but, if the ring of the coefficients is not a field, there exist non-free modules.

Given any set S, there is a free module with basis S, which is called free module on S or module of formal linear combinations of the elements of S.

Definition

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

For an ${\displaystyle R}$-module ${\displaystyle M}$, the set ${\displaystyle E\subseteq M}$ is a basis for ${\displaystyle M}$ if:

1. ${\displaystyle E}$ is a generating set for ${\displaystyle M}$; that is to say, every element of ${\displaystyle M}$ is a finite sum of elements of ${\displaystyle E}$ multiplied by coefficients in ${\displaystyle R}$;
2. ${\displaystyle E}$ is linearly independent, that is, ${\displaystyle r_{1}e_{1}+r_{2}e_{2}+\cdots +r_{n}e_{n}=0_{M}}$ for ${\displaystyle e_{1},e_{2},\ldots ,e_{n}}$ distinct elements of ${\displaystyle E}$ implies that ${\displaystyle r_{1}=r_{2}=\cdots =r_{n}=0_{R}}$ (where ${\displaystyle 0_{M}}$ is the zero element of ${\displaystyle M}$ and ${\displaystyle 0_{R}}$ is the zero element of ${\displaystyle R}$).

If ${\displaystyle R}$ 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 ${\displaystyle M}$. The free module 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 ${\displaystyle x\in M}$.

Formal linear combinations

Given a set E, we can construct a free R-module that has E as a basis. This module is called the module of the formal linear combinations of elements of E, or the free module over E, and is denoted R(E).

Given a finite subset {X1, ..., Xn} of E, a formal linear combination of X1, ..., Xn is an expression

a1X1 + ··· + anXn,

where the ai belong to R.

If some ai is zero, the formal linear combination is identified (that is, considered as equal) with the formal linear combination in which the corresponding summand is removed. Similarly, a summand Xi is simplified as Xi.

With these identifications, it is straightforward to show that all formal linear combinations of elements of E form a free module, which has E as a basis.

The formal linear combinations of a single element X are simply the products aX with a in R. They form a module that is isomorphic to R. It follows that the module R(E) of all linear combinations of the elements of E may be considered as the direct sum of ${\displaystyle |E|}$ copies of R.

Another construction

The module R(E) may also be constructed in the following equivalent (that is isomorphic) way.

Let us consider the set C(E) of the functions ${\displaystyle f:E\to R}$ such that f(x) = 0 for all but finitely many x in E. This set has a structure of a module if the addition is defined by

${\displaystyle (f+g)(x)=f(x)+g(x),\quad \forall x\in E,}$

and the scalar multiplication by

${\displaystyle (af)(x)=a(f(x)),\quad \forall x\in E.}$

A basis of C(E) consists of the functions ${\displaystyle \delta _{a}}$ that have the value zero for all entries, except one, for which the value is one:

${\displaystyle \delta _{a}(x)={\begin{cases}1\quad {\mbox{if }}x=a\\0\quad {\mbox{if }}x\neq a\end{cases}}}$

(this is a variant of the Kronecker delta, and a particular case of the indicator function for the set {a}). This basis is commonly called the canonical basis.

The mapping ${\displaystyle a\mapsto \delta _{a}}$ is a bijection between E and this canonical basis. It induces a canonical isomorphism between the module of the formal linear combinations and C(E), which allows us to identify these two free modules.

Universal property

The inclusion mapping ${\displaystyle \iota :E\to R^{(E)}}$ defined above is universal in the following sense. Given an arbitrary mapping ${\displaystyle \varphi :E\to M}$ from a set E into a R-module M, there exists a unique module homomorphism ${\displaystyle \psi :R^{(E)}\to M}$ such that ${\displaystyle \varphi =\psi \circ \iota }$.

As usual for universal properties, this defines R(E) up to a canonical isomorphism. Also the mapping ${\displaystyle \iota :E\to R^{(E)}}$ may naturally be extended into a functor from the category of sets into the category of R-modules. This functor is a left adjoint of the forgetful functor that maps a module to its underlying set.

Generalizations

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.