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 subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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 Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

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, and this multiplication is associative (when used with the multiplication in the ring) and distributive. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

Modules are very closely related to the representation theory of groups. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic

Contents 1 Motivation 2 Formal definition 3 Examples 4 Submodules and homomorphisms 5 Types of modules 6 Relation to representation theory 7 Generalizations 8 See also 9 References

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 Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

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. Mathematicians (and those in related sciences very frequently speak of whether a mathematical object &mdash a Number, a function, a set, a space In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i 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). Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction In Mathematics, the invariant basis number (IBN property of a ring R is the property that all free modules over R are similarly 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. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the e. as rx for r in R and x in M) such that

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). In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Ring theory or 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 object's endomorphism ring, which encodes several internal properties of the object Thus a module is a ring action on an abelian group (cf. group action). In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In this sense, module theory generalizes representation theory, which deals with group actions on vector spaces, or equivalently group ring actions. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively

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 Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In this article however, all rings and modules are assumed to be unital.

A bimodule is a module which is a left module and a right module such that the two multiplications 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 has the commutative property

Examples

• If K is a field, then the concepts "K-vector space" (a vector space over K) and K-module are identical. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added
• 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. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French 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. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. 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. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction
• 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 Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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). A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, the real numbers may be described informally in several different ways 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. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is 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. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space In Mathematics, particularly in Abstract algebra and Homological algebra, the concept of projective module over a ring R is a more flexible generalisation In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are
• 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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix
• If R is any ring and I is any left ideal in R, then I is a left module over R. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. 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.

Submodules and homomorphisms

Suppose M is a left R-module and N is a subgroup of M. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of 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).

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, N₁, N₂ of M such that N₁ ⊂ N₂, then the two submodules are equal: (N₁ + U) ∩ N₂ = N₁ + (U ∩ N₂). In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' In the branch of mathematics called Order theory, a modular lattice is a lattice that satisfies the following self-dual condition Modular law: x

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,

f(rm + sn) = rf(m) + sf(n). In Mathematics and related technical fields the term map or mapping is often a Synonym for function.

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

A bijective module homomorphism is an isomorphism of modules, and the two modules are called isomorphic. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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. 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 The isomorphism theorems familiar from groups and vector spaces are also valid for R-modules. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural

The left R-modules, together with their module homomorphisms, form a category, written as R-Mod. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets This is an abelian category. In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist

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, 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 over a ring 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. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction 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. In Mathematics, particularly in Abstract algebra and Homological algebra, the concept of projective module over a ring R is a more flexible generalisation The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

Injective. Injective modules are defined dually to projective modules. In Mathematics, especially in the area of Abstract algebra known as Module theory, an injective module is a module Q that shares certain

Simple. A simple module S is a module that is not {0} and whose only submodules are {0} and S. In Abstract algebra, a (left or right module S over a ring R is called simple or irreducible if it is not the Zero Simple modules are sometimes called irreducible.

Indecomposable. An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. In Abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a Direct sum of two non-zero Submodules Indecomposable The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction Every simple module is indecomposable.

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. In Mathematics, specifically Module theory, annihilators are a concept that formalizes torsion and generalizes torsion and Orthogonal complement

Noetherian. A noetherian module is a module such that every submodule is finitely generated. In Abstract algebra, an Noetherian module is a module that satisfies the Ascending chain condition on its Submodules where the submodules are Equivalently, every increasing chain of submodules becomes stationary after finitely many steps.

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 M = ⊕_x M_x over a graded ring R = ⊕_x R_x such that R_xM_y ⊂ M_{x + y} 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 In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure

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,+). In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function 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). In Ring theory or 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. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers

Generalizations

Any ring R can be viewed as a preadditive category with a single object. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category With this understanding, a left R-module is nothing but a (covariant) additive functor from R to the category Ab of abelian groups. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category 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. In Category theory, a branch of Mathematics, the Functors between two given categories can themselves be turned into a category the morphisms in this functor

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. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. These form a category O_X-Mod, and play an important role in the scheme-theoretic approach to algebraic geometry. 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 techniques of Abstract algebra, especially Commutative algebra, with 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. In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse Modules over rings are abelian groups, but modules over semirings are only commutative monoids. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation 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). In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse This allows a further generalization of the concept of vector space incorporating the semirings from theoretical computer science. 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

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

References

• F. In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively 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 In the branch of Mathematical logic called Model theory, an elementary class (or axiomatizable class) is a class consisting of all 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

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