Generally, something that is **modular** is constructed so as to facilitate easy assembly, flexible arrangement, and/or repair.

For module in Linux see module (linux)

In abstract algebra, a **left R-module** consists of an abelian group (

For all *r*,*s* in *R*, *x*,*y* in *M*, we have

- (
*rs*)*x*=*r*(*sx*) - (
*r*+*s*)*x*=*rx*+*sx* -
*r*(*x*+*y*) =*rx*+*ry* - 1
*x*=*x*

Some authors omit condition 4 for the general definition of left modules, and call the above defined structures "unital left modules". In this encyclopedia however, all modules are assumed to be unital.

A **right R-module**

If *R* is a field, then an *R*-module is also called a vector space. Modules are thus generalizations of vector spaces, and much of the theory of modules consists of recovering desirable properties of vector spaces in the realm of modules over certain rings. However, in general, an *R*-module may not have a basis.

Table of contents |

2 Submodules and homomorphisms 3 Alternative definition as representations |

- Every abelian group
*M*is a module over the ring of integers**Z**if we define*nx*=*x*+*x*+ ... +*x*(*n*summands) for*n*> 0, 0*x*= 0, and (-*n*)*x*= -(*nx*) for*n*< 0. - If
*R*is any ring and*n*a natural number, then the cartesian product*R*^{n}is a module over*R*if we use the component-wise operations. - If
*M*is a smooth manifold, then the smooth functions from*M*to the real numbers form a ring*R*. The set of all vector fields defined on*M*form a module over*R*, and so do the tensor fields and the differential forms on*M*. - The square
*n*-by-*n*matrices with real entries form a ring*R*, and the Euclidean space**R**^{n}is a left module over this ring if we define the module operation via matrix multiplication. - If
*R*is any ring and*I*is any left ideal in*R*, then*I*is a left module over*R*.

Suppose *M* is an *R*-module and *N* is a subgroup
of *M*. Then *N* is a **submodule** (or *R*-submodule, to be more explicit) if, for any *n* in *N* and any *r* in *R*, the product *rn* is in *N* (or *nr* for a right module).

If *M* and *N* are left *R*-modules, then a map
*f* : *M* `->` *N* is a **homomorphism of R-modules** if, for any

If *M* is a left *R*-module, then the *action* of an element *r* in *R* is defined to be the map *M* → *M* that sends each *x* to *rx* (or *xr* in the case of a right module), and is necessarily a group endomorphism of the abelian group (*M*,+). The set of all group endomorphisms of *M* is denoted End_{Z}(*M*) and forms a ring under addition and composition, and sending a ring element *r* of *R* to its action actually defines a ring homomorphism from *R* to End_{Z}(*M*).

Such a ring homorphism *R* → End_{Z}(*M*) is called a *representation* of *R* over the abelian group *M*; an alternative and equivalent way of defining left *R*-modules is to say that a left *R*-module is an abelian group *M* together with a representation of *R* over it.

A representation is called *faithful* if and only if the map *R* → End_{Z}(*M*) is injective. In terms of modules, this means that if *r* is an element of *R* such that *rx*=0 for all *x* in *M*, then *r*=0. Every abelian group is a faithful module over the integers or over some modular arithmetic **Z**/*n***Z**.