In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Thus, a module, like a vector space, is an abelian group; a product is defined between elements of the ring and elements of the module, and this multiplication is associative (when used with the multiplication in the ring) and distributive. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

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 mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... This picture illustrates how the hours in a clock form a group. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...

Motivation

In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

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 "well-behaved" 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 basis whose cardinality is then unique (assuming the axiom of choice). Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... In mathematics, a free module is a module having a free basis. ... in mathematics, the invariant basis number (IBN) property of a ring R is the property that all free modules over R are similarly well-behaved to vector spaces with respect to the uniqueness of their ranks. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

Formal definition

A left R-module over the ring R consists of an abelian group (M, +) and an operation R × M → M (called scalar multiplication, usually just written by juxtaposition, i.e. as rx for r in R and x in M) such that In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

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

1. r(x+y) = rx+ry
2. (r+s)x = rx+sx
3. (rs)x = r(sx)
4. 1x = x

If one writes the scalar action as f_r so that f_r(x) = rx, and f for the map which 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 f is a ring homomorphism from R to the endomorphism ring End(M). Thus a module is a ring action on an abelian group (cf. group action). In this sense, module theory generalizes representation theory, which deals with group actions on vector spaces, or equivalently group ring actions. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described...

Usually, we simply write "a left R-module M" or _RM. A right R-module M or M_R is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of 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 in the above definition, and call the above structures "unital left modules". In this article however, all rings and modules are assumed to be unital. In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ...

A bimodule is a module which is a left module and a right module such that the two mutiplications are compatible. In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. ...

If R is commutative, then left R-modules are the same as right R-modules and are simply called R-modules. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

Examples

• If K is a field, then the concepts "K-vector space" and K-module are identical.
• The concept of a Z-module 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).
• If R is any ring and n a natural number, then the cartesian product Rⁿ is both a left and a right module over R if we use the component-wise operations. Hence when n = 1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial R-module {0} consisting only of its identity element. Modules of this type are called free and the number n is then the rank of the free module.
• If S is a nonempty set, M is a left R-module, 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 R-module. The right R-module case is analogous. In particular, if R is commutative then the collection of R-module homomorphisms h : M → N (see below) is an R-module (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 n-by-n matrices with real entries form a ring R, and the Euclidean space Rⁿ 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 R-module 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.

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... The integers are commonly denoted by the above symbol. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, the Cartesian product is a direct product of sets. ... In mathematics, a free module is a module having a free basis. ... Rank means a wide variety of things in mathematics, including: Rank (linear algebra) Rank of a tensor Rank of an array Rank of an abelian group Rank (set theory) Rank-into-rank Rank of a greedoid This is a disambiguation page — a navigational aid which lists other pages that... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... Partial plot of a function f. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ... Swans theorem relates vector bundles to projective modules and gives rise to a common intuition throughout mathematics: projective modules over commutative rings are like vector bundles on compact spaces. Differential geometry Suppose M is a compact C∞-manifold, and a smooth vector bundle V is given on M... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... This article gives an overview of the various ways to perform matrix multiplication. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

Submodules and homomorphisms

Suppose M is a left 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). In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

If M and N are left R-modules, then a map f : M → N is a homomorphism of R-modules if, for any m, n in M and r, s in R, In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...

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. In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ...

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. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

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 abelian groups and vector spaces are also valid for R-modules. In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ...

The left R-modules, together with their module homomorphisms, form a category, written as R-Mod. This is an abelian category. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...

Types of modules

Finitely generated. A module M is finitely generated if there exist finitely many elements x₁,...,x_n in M such that every element of M is a linear combination of those elements with coefficients from the scalar ring R. In mathematics, a module is a finitely-generated module if it has a finite generating set. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

Cyclic module. A module is called cyclic module if it is generated by one element. In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module which is generated by one element. ...

Free. A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring R. These are the modules that behave very much like vector spaces. In mathematics, a free module is a module having a free basis. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

Projective. Projective modules are direct summands of free modules and share many of their desirable properties. In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

Injective. Injective modules are defined dually to projective modules. In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ...

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. In abstract algebra, a (left or right) module S over a ring R is called simple if it is not the zero module and if its only submodules are 0 and S. Understanding the simple modules over a ring is usually helpful because they form the building blocks of all...

Indecomposable. An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable. In abstract algebra, a module is defined to be indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

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. Annihilators are a concept that occurs in ring theory, a branch of mathematics. ...

Noetherian. A noetherian module is a module such that every submodule is finitely generated. Equivalently, every increasing chain of submodules becomes stationary after finitely many steps. In ring theory, if R is a ring and M is a module over R, then M is Noetherian if M satisfies the ascending chain condition on its submodules when they are ordered by inclusion. ...

Artinian. An artinian module is a module in which every decreasing chain of submodules becomes stationary after finitely many steps. In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its submodules. ...

Graded. A graded module is a module decomposable as a direct sum over a graded ring such that for all x and y. In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ...

Relation to representation theory

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). Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

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 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/nZ. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... The integers are commonly denoted by the above symbol. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

Generalizations

Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left R-module is nothing but a (covariant) additive functor from R to the category Ab of abelian groups. Right R-modules 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 C-Mod which is the natural generalization of the module category R-Mod. A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ...

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 scheme-theoretic 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). In mathematics, a ringed space is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

One can also consider modules over a semiring. Then the module is only a commutative monoid. 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. In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

See also

• vector space
• group ring
• algebra (ring theory)
• module (model theory)

In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described... In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ... In mathematics, an axiomatizable class is a class of mathematical structures which are all models of a fixed set of sentences in formal (typically first order) logic. ...

References

• F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, ISBN 0-387-97845-3, ISBN 3-540-97845-3