This article includes a list of general references, but it remains largely unverified because it lacks sufficient corresponding inline citations. (May 2015) (Learn how and when to remove this template message) 
Algebraic structures 

Algebraic structure → Ring theory Ring theory 

Basic concepts Rings
Related structures

Commutative rings
padic number theory and decimals

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring (with identity) and a multiplication (on the left and/or on the right) is defined between elements of the ring and elements of the module. A module taking its scalars from a ring R is called an Rmodule.
Thus, a module, like a vector space, is an additive abelian group; a product is defined between elements of the ring and elements of the module that is distributive over the addition operation of each parameter and is compatible with the ring multiplication.
Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology.
In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In noncommutative algebra the distinction between left ideals, ideals, and modules becomes more pronounced, though some ringtheoretic conditions can be expressed either about left ideals or left modules.
Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "wellbehaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do, free modules, need not have a unique rank if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the axiom of choice in general, but not in the case of finitedimensional spaces, or certain wellbehaved infinitedimensional spaces such as L^{p} spaces.)
Suppose that R is a ring and 1_{R} is its multiplicative identity. A left Rmodule M consists of an abelian group (M, +) and an operation ⋅ : R × M → M such that for all r, s in R and x, y in M, we have:
The operation of the ring on M is called scalar multiplication, and is usually written by juxtaposition, i.e. as rx for r in R and x in M, though here it is denoted as r ⋅ x to distinguish it from the ring multiplication operation, denoted here by juxtaposition. The notation _{R}M indicates a left Rmodule M. A right Rmodule M or M_{R} is defined similarly, except that the ring acts on the right; i.e., scalar multiplication takes the form ⋅ : M × R → M, and the above axioms are written with scalars r and s on the right of x and y.
Authors who do not require rings to be unital omit condition 4 above in the definition of an Rmodule, and so would call the structures defined above "unital left Rmodules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be unital.^{[1]}
If one writes the scalar action as f_{r} so that f_{r}(x) = r ⋅ x, and f for the map that takes each r to its corresponding map f_{r} , then the first axiom states that every f_{r} is a group endomorphism of M, and the other three axioms assert that the map f : R → End(M) given by r ↦ f_{r} is a ring homomorphism from R to the endomorphism ring End(M).^{[2]} Thus a module is a ring action on an abelian group (cf. group action. Also consider monoid action of multiplicative structure of R). In this sense, module theory generalizes representation theory, which deals with group actions on vector spaces, or equivalently group ring actions.
A bimodule is a module that is a left module and a right module such that the two multiplications are compatible.
If R is commutative, then left Rmodules are the same as right Rmodules and are simply called Rmodules.
Suppose M is a left Rmodule and N is a subgroup of M. Then N is a submodule (or more explicitly an Rsubmodule) if for any n in N and any r in R, the product r ⋅ n is in N (or n ⋅ r for a right Rmodule).
If X is any subset of an Rmodule, then the submodule spanned by X is defined to be where N runs over the submodules of M which contains X, or explicitly , which is important in the definition of tensor products.^{[3]}
The set of submodules of a given module M, together with the two binary operations + and ∩, forms a lattice which satisfies the modular law: Given submodules U, N_{1}, N_{2} of M such that N_{1} ⊂ N_{2}, then the following two submodules are equal: (N_{1} + U) ∩ N_{2} = N_{1} + (U ∩ N_{2}).
If M and N are left Rmodules, then a map f : M → N is a homomorphism of Rmodules if for any m, n in M and r, s in R,
This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects. Another name for a homomorphism of Rmodules is an Rlinear map.
A bijective module homomorphism is an isomorphism of modules, and the two modules are called isomorphic. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.
The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f, and the image of f is the submodule of N consisting of values f(m) on all elements m of M.^{[4]} The isomorphism theorems familiar from groups and vector spaces are also valid for Rmodules.
Given a ring R, the set of all left Rmodules together with their module homomorphisms forms an abelian category, denoted by RMod (see category of modules).
A representation of a group G over a field k is a module over the group ring k[G].
If M is a left Rmodule, 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 homomorphism R → End_{Z}(M) is called a representation of R over the abelian group M; an alternative and equivalent way of defining left Rmodules is to say that a left Rmodule 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/nZ.
Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left Rmodule is just a covariant additive functor from R to the category Ab of abelian groups, and right Rmodules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C. These functors form a functor category CMod which is the natural generalization of the module category RMod.
Modules over commutative rings can be generalized in a different direction: take a ringed space (X, O_{X}) and consider the sheaves of O_{X}modules (see sheaf of modules). These form a category O_{X}Mod, and play an important role in modern algebraic geometry. If X has only a single point, then this is a module category in the old sense over the commutative ring O_{X}(X).
One can also consider modules over a semiring. Modules over rings are abelian groups, but modules over semirings are only commutative monoids. Most applications of modules are still possible. In particular, for any semiring S, the matrices over S form a semiring over which the tuples of elements from S are a module (in this generalized sense only). This allows a further generalization of the concept of vector space incorporating the semirings from theoretical computer science.
Over nearrings, one can consider nearring modules, a nonabelian generalization of modules.^{[citation needed]}