Free module

Template:Short description In mathematics, a free module is a module that has a basis, that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.

Given any set Template:Math and ring Template:Math, there is a free Template:Math-module with basis Template:Math, which is called the free module on Template:Math or module of formal Template:Math-linear combinations of the elements of Template:Math.

A free abelian group is precisely a free module over the ring $\mathbb{Z}$ of integers.

Definition

For a ring $R$ and 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 $R$; and
• $E$ is linearly independent if for every $\{e_1,\dots ,e_n\}\subset E$ of distinct elements, $r_1{e}_1+r_2{e}_2+\cdots +r_n{e}_n=0_M$ 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 $R$).

A free module is a module with a basis.

An immediate consequence of the second half of the definition is that the coefficients in the first half are unique for each element of M.

If $R$ has invariant basis number, then by definition any two bases have the same cardinality. For example, nonzero commutative rings have invariant basis number. The cardinality of any (and therefore every) basis is called the rank of the free module $M$. If this cardinality is finite, the free module is said to be free of finite rank, or free of rank Template:Mvar if the rank is known to be Template:Mvar.

Examples

Let R be a ring.

• R is a free module of rank one over itself (either as a left or right module); any unit element is a basis.
• More generally, If R is commutative, a nonzero ideal I of R is free if and only if it is a principal ideal generated by a nonzerodivisor, with a generator being a basis.
• Over a principal ideal domain (e.g., $\mathbb{Z}$), a submodule of a free module is free.
• If R is commutative, the polynomial ring $R[X]$ in indeterminate X is a free module with a possible basis 1, X, X², ....
• Let $A[t]$ be a polynomial ring over a commutative ring A, f a monic polynomial of degree d there, $B=A[t]/(f)$ and $\xi$ the image of t in B. Then B contains A as a subring and is free as an A-module with a basis $1,\xi ,\dots ,{\xi }^{d-1}$.
• For any non-negative integer n, $R^n=R\times \cdots \times R$, the cartesian product of n copies of R as a left R-module, is free. If R has invariant basis number, then its rank is n.
• A direct sum of free modules is free, while an infinite cartesian product of free modules is generally not free (cf. the Baer–Specker group).
• A finitely generated module over a commutative local ring is free if and only if it is faithfully flat. Also, Kaplansky's theorem states a projective module over a (possibly non-commutative) local ring is free.
• Sometimes, whether a module is free or not is undecidable in the set-theoretic sense. A famous example is the Whitehead problem, which asks whether a Whitehead group is free or not. As it turns out, the problem is independent of ZFC.

Formal linear combinations

Template:AnchorGiven a set Template:Math and ring Template:Math, there is a free Template:Math-module that has Template:Math as a basis: namely, the direct sum of copies of R indexed by E

$R^{(E)}=\underset{e\in E}{⨁}R$.

Explicitly, it is the submodule of the Cartesian product $\prod _{E}R$ (R is viewed as say a left module) that consists of the elements that have only finitely many nonzero components. One can embed E into Template:Math as a subset by identifying an element e with that of Template:Math whose e-th component is 1 (the unity of R) and all the other components are zero. Then each element of Template:Math can be written uniquely as

$\sum _{e\in E}{c}_{e}e,$

where only finitely many $c_e$ are nonzero. It is called a formal linear combination of elements of Template:Math.

A similar argument shows that every free left (resp. right) R-module is isomorphic to a direct sum of copies of R as left (resp. right) module.

Another construction

The free module Template:Math may also be constructed in the following equivalent way.

Given a ring R and a set E, first as a set we let

$R^{(E)}=\{f:E\to R\mid f(x)=0\text{ for all but finitely many }x\in E\}.$

We equip it with a structure of a left module such that the addition is defined by: for x in E,

$(f+g)(x)=f(x)+g(x)$

and the scalar multiplication by: for r in R and x in E,

$(rf)(x)=rf(x)$

Now, as an R-valued function on E, each f in $R^{(E)}$ can be written uniquely as

$f=\sum _{e\in E}{c}_e{\delta }_e$

where $c_e$ are in R and only finitely many of them are nonzero and ${\delta }_e$ is given as

${\delta }_e(x)=\{\begin{array}{c}{1}_R\text{if }x=e\\ 0_R\text{if }x\ne e\end{array}$

(this is a variant of the Kronecker delta). The above means that the subset $\{{\delta }_e\mid e\in E\}$ of $R^{(E)}$ is a basis of $R^{(E)}$. The mapping $e\mapsto {\delta }_e$ is a bijection between Template:Math and this basis. Through this bijection, $R^{(E)}$ is a free module with the basis E.

Universal property

The inclusion mapping $\iota :E\to R^{(E)}$ defined above is universal in the following sense. Given an arbitrary function $f:E\to N$ from a set Template:Math to a left Template:Math-module Template:Math, there exists a unique module homomorphism $\bar{f}:R^{(E)}\to N$ such that $f=\bar{f}\circ \iota$; namely, $\bar{f}$ is defined by the formula:

$\bar{f}\left(\sum _{e\in E}{r}_{e}e\right)=\sum _{e\in E}{r}_{e}f(e)$

and $\bar{f}$ is said to be obtained by extending $f$ by linearity. The uniqueness means that each R-linear map $R^{(E)}\to N$ is uniquely determined by its restriction to E.

As usual for universal properties, this defines Template:Math up to a canonical isomorphism. Also the formation of $\iota :E\to R^{(E)}$ for each set E determines a functor

$R^{(-)}:\mathbf{\text{Set}}\to R\text{-}𝖬𝗈𝖽,E\mapsto R^{(E)}$,

from the category of sets to the category of left Template:Math-modules. It is called the free functor and satisfies a natural relation: for each set E and a left module N,

${\mathrm{Hom}}_{\mathbf{\text{Set}}}(E,U(N))\simeq {\mathrm{Hom}}_R(R^{(E)},N),f\mapsto \bar{f}$

where $U:R\text{-}𝖬𝗈𝖽\to \mathbf{\text{Set}}$ is the forgetful functor, meaning $R^{(-)}$ is a left adjoint of the forgetful functor.

Generalizations

Many statements true for free modules extend to certain larger classes of modules. Projective modules are direct summands of free modules. Flat modules are defined by the property that tensoring with them preserves exact sequences. Torsion-free modules form an even broader class. For a finitely generated module over a PID (such as Z), the properties free, projective, flat, and torsion-free are equivalent.

Module properties in commutative algebra

See local ring, perfect ring and Dedekind ring.

See also

• Free object
• Projective object
• free presentation
• free resolution
• Quillen–Suslin theorem
• stably free module
• generic freeness

References

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