In abstract algebra, the concept of 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 ring. Modules also generalize the notion of abelian groups, which are modules over the ring of integers.
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 both parameters 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.
Introduction
Motivation
In a vector space, the set of scalars forms 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 ring theoretic conditions can be expressed either about left ideals or left modules.
Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces 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.)
Formal definition
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:

r(x+y) = rx + ry

(r+s)x = rx + sx

(rs)x = r(sx)

1_Rx = x.
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. 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.
If one writes the scalar action as f_{r} so that f_{r}(x) = rx, 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 homomorphism 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).^{[1]} 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.
Examples

If K is a field, then the concepts "Kvector space" (a vector space over K) and Kmodule are identical.

The concept of a Zmodule agrees with the notion of an abelian group. That is, every abelian group is a module over the ring of integers Z in a unique way. For n > 0, let nx = x + x + ... + x (n summands), 0x = 0, and (−n)x = −(nx). Such a module need not have a basis—groups containing torsion elements do not. (For example, in the group of integers modulo 3, one cannot find even one element which satisfies the definition of a linearly independent set since when an integer such as 3 or 6 multiplies an element the result is 0. However if a finite field is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)

If R is any ring and n a natural number, then the cartesian product R^{n} is both a left and a right module over R if we use the componentwise operations. Hence when n = 1, R is an Rmodule, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial Rmodule {0} consisting only of its identity element. Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module.

If S is a nonempty set, M is a left Rmodule, and M^{S} is the collection of all functions f : S → M, then with addition and scalar multiplication in M^{S} defined by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(s), M^{S} is a left Rmodule. The right Rmodule case is analogous. In particular, if R is commutative then the collection of Rmodule homomorphisms h : M → N (see below) is an Rmodule (and in fact a submodule of N^{M}).

If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C^{∞}(X). The set of all smooth vector fields defined on X form a module over C^{∞}(X), and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over C^{∞}(X), and by Swan's theorem, every projective module is isomorphic to the module of sections of some bundle; the category of C^{∞}(X)modules and the category of vector bundles over X are equivalent.

The square nbyn 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. Analogously of course, right ideals are right modules.

If R is a ring, we can define the ring R^{op} which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in R^{op}. Any left Rmodule M can then be seen to be a right module over R^{op}, and any right module over R can be considered a left module over R^{op}.

There are modules of a Lie algebra as well.
Submodules and homomorphisms
Suppose M is a left Rmodule and N is a subgroup of M. Then N is a submodule (or Rsubmodule, 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).
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,

f(rm + sn) = rf(m) + sf(n)
This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects. Another name for a homomorphism of modules over R 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. The isomorphism theorems familiar from groups and vector spaces are also valid for Rmodules.
The left Rmodules, together with their module homomorphisms, form a category, written as RMod. This is an abelian category.
Types of modules
Finitely generated. An Rmodule M is finitely generated if there exist finitely many elements x_{1},...,x_{n} in M such that every element of M is a linear combination of those elements with coefficients from the ring R.
Cyclic. A module is called a cyclic module if it is generated by one element.
Free. A free Rmodule is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces.
Projective. Projective modules are direct summands of free modules and share many of their desirable properties.
Injective. Injective modules are defined dually to projective modules.
Flat. A module is called flat if taking the tensor product of it with any exact sequence of Rmodules preserves exactness.
Torsionless module. A module is called torsionless if it embeds into its algebraic dual.
Simple. A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible.^{[2]}
Semisimple. A semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called completely reducible.
Indecomposable. An indecomposable module is a nonzero module that cannot be written as a direct sum of two nonzero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. uniform modules).
Faithful. A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. rx ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal.
Torsionfree. A torsionfree module is a module over a ring such that 0 is the only element annihilated by a regular element (non zerodivisor) of the ring.
Noetherian. A Noetherian module is a module which satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.
Artinian. An Artinian module is a module which satisfies the descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.
Graded. A graded module is a module with a decomposition as a direct sum M = ⨁_{x} M_{x} over a graded ring R = ⨁_{x} R_{x} such that R_{x}M_{y} ⊂ M_{x + y} for all x and y.
Uniform. A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.
Further notions
Relation to representation theory
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.
Generalizations
Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left Rmodule is nothing but a (covariant) additive functor from R to the category Ab of abelian groups. 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. These form a category O_{X}Mod, and play an important role in the schemetheoretic approach to 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.
See also
Notes

^ This is the endomorphism ring of the additive group M. If R is commutative, then these endomorphisms are additionally R linear.

^ Jacobson (1964), p. 4, Def. 1; Irreducible Module at PlanetMath.org.
References

F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., SpringerVerlag, New York, 1992, ISBN 0387978453, ISBN 3540978453

Nathan Jacobson. Structure of rings. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964, ISBN 9780821810378
External links
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