Module

A module is a type of abstract algebraic structure related to both vector spaces and abelian groups.

Idea
There are multiple ways to think of modules as generalizations of other familiar objects.

Vector spaces
The most common way to think of modules is as a generalization of the concept of a vector space. While a vector space requires its scalars to form a field, that is every scalar has an inverse, a module relaxes that restriction.

Abelian groups
In group theory it is common to see an element of a group $a$ raised to the power of some integer $n$.

$$ a^n $$

This represents combining copies of $a$ (or if $n$ is negative, then $$\mathrm{inv}(a)$$) in the same fashion that raising an integer to an integer power combines copies of that integer.

This operation has a few nice properties, but in an abelian group specifically this action distributes:

$$ (ab)^3=ababab=aaabbb=a^3b^3 $$

Thus there is a special relationship between abelian groups and the ring of integers. A module generalizes this idea, so that we have an operation that behaves like this power operation but instead of being between an group and the ring of integers it is between a group and any ring. Abelian groups correspond one to one with -modules.

Lattices
Lattices are of particular interest since they are relevant to the study of polytopes. A lattice is a particular kind of abelian group associated with a vector space. Lattices generally do not form a proper vector space themselves, however they do have vector-space-like properties not usually associated with abelian groups. By generalizing both vector spaces and abelian groups to modules we can use some of the language of vector spaces to talk about lattices. For example lattices as turn out to be free-modules and can be given a basis and dimension.

Group actions
Another way to think about modules is as being the idea of group actions applied to rings.

Note that a group action can be defined as a group homomorphism from a group to an automorphism group:

$$ G \rightarrow \mathrm{Aut}(X) $$

Similarly a module can be defined as a ring homomorphism from a ring to an endomorphism ring:

$$ R \rightarrow \mathrm{End}(X) $$

Definitions
Modules are defined in respect to a ring $R$. When that ring, $R$, is noncommutative, there exists distinct left and right modules, which have similar yet distinct definitions. When $R$ is commutative left modules are right modules and so no distinction is made.

Algebraic definition
For some ring $R$ and abelian group $G$, a left $R$-module is a function $$* : R \times G \rightarrow G$$ such that:


 * $$(f \times_R g) * v = f * (g * v)$$
 * $$f * (v +_G w) = (f * v) +_G (f * w)$$
 * $$(f +_R g) * v = (f * v) +_G (g * v)$$
 * $$1_R * v = v$$

In a right $R$-module the first law is replaced with


 * $$(f \times_R g) * v = g * (f * v)$$

Homomorphism definition
A left $R$-module is a ring homomorphism from $R$ to the endomorphism ring of an abelian group $G$, $$\rho : R \rightarrow \mathrm{End}(G)$$.

A right $R$-module is a ring homomorphism from $R$ to $$\mathrm{Op}(\mathrm{End}(G))$$, where $$\mathrm{Op}$$ is the opposite ring, that is the same ring with its multiplicative action flipped. Alternatively a right $R$-module can be though of as a left $$\mathrm{Op}(R)$$-module.

Categorical definition
In category theory a ring, $R$, is simply a preadditive category with one object. A left $R$-module is thus a covariant additive functor from $R$ to $$\mathrm{Ab}$$ the category of abelian groups. A right $R$-module is a contravariant functor.

Linear independence
For an $R$-module on an abelian group $G$, a sequence of elements in $G$, $$v_0^n$$ is linearly dependent iff there exists some sequence of scalars of $R$, $$a_0^n$$ such that, at least one of the scalars is non-zero, and:

$$ 0 = \sum_{i=0}^na_i*v_i $$

Where $0$ is the identity of $G$.

A sequence of elements of $G$, which is not linearly dependent is called linearly independent. It is common to talk about the linear independence of a set of elements rather than a specific sequence.

The notion of linear dependence can be sometimes counter intuitive. For example for any non-trivial $R$-module the set $$\{0\}$$ is linearly dependent, even though it contains only one element. This is also true in vector spaces, but in modules this counter intuition extends even farther. For example if we have the -module over the abelian group $C_{3}$ (the cyclic group of size 3), then the set $$\{1\}$$ is not linearly independent, since $$3*1=1+1+1=0$$. In fact the only linearly independent set on this module is $$\{\}$$.

Basis
The concept of basis can be extended to modules without modification. A basis of a module is a linearly independent set of non-zero vectors such that its span is module. However unlike vector spaces, not every module has a basis.

Modules that do have a basis are called free-modules. The bases of a module are not in general all of the same size.

Direct product
As with vector spaces a direct product can be defined for modules. The direct product of two $R$-modules $M$ and $N$ is a $R$-module with its abelian group being the direct product of the ablian groups of $M$ and $N$ and its scalar multiplication being defined as:

$$ a*_{M\times N}(v,w) = (a*_Mv,a*_Nw) $$

If $M$ has a basis of size $m$ and $N$ has a basis of size $n$, then there is a basis of $$M\times N$$ of size $$m+n$$.

Examples

 * Since the set of integers modulo 5 form a abelian group they also form a -module. However since $$a^5 = 0$$ for any $a$, no element can be a part of a linearly independent set.  Thus the only linearly independent set is the empty set.  The empty set does not span the module, thus this is a module with no basis.
 * Vectors with integer coordinates form a -module rather than a vector space because is not a field.  This makes them an abelian group.
 * Tilings of a vector space form modules in that space. Particularly a isotopic tiling will form a -module with each tile being represented by the coordinate of its center. This subspace is not a vector space, but it is a free module and has a definite dimension.