Simple module
Encyclopedia
In mathematics
, specifically in ring theory
, the simple modules over a ring R are the (left or right) module
s over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if
every cyclic submodule
generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length
, and they are analogous to the simple group
s in group theory.
In this article, all modules will be assumed to be right unital modules over a ring R.
-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups. These are the cyclic group
s of prime
order
.
If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal. Conversely, if I is not minimal, then there is a non-zero right ideal J properly contained in I. J is a right submodule of I, so I is not simple.
If I is a right ideal of R, then R/I is simple if and only I is a maximal right ideal: If M is a non-zero proper submodule of R/I, then the preimage of M under the quotient map is a right ideal which is not equal to R and which properly contains I. Therefore I is not maximal. Conversely, if I is not maximal, then there is a right ideal J properly containing I. The quotient map has a non-zero kernel which is not equal to , and therefore is not simple.
Every simple R-module is isomorphic to a quotient R/m where m is a maximal right ideal
of R. By the above paragraph, any quotient R/m is a simple module. Conversely, suppose that M is a simple R-module. Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x. The statement that xR = M is equivalent to the surjectivity of the homomorphism that sends r to xr. The kernel of this homomorphism is a right ideal I of R, and a standard theorem states that M is isomorphic to R/I. By the above paragraph, we find that I is a maximal right ideal. Therefore M is isomorphic to a quotient of R by a maximal right ideal.
If k is a field
and G is a group
, then a group representation
of G is a left module over the group ring
kG. The simple kG modules are also known as irreducible representations. A major aim of representation theory
is to understand the irreducible representations of groups.
1; this is a reformulation of the definition.
Every simple module is indecomposable
, but the converse is in general not true.
Every simple module is cyclic
, that is it is generated by one element.
Not every module has a simple submodule; consider for instance the Z-module Z in light of the first example above.
Let M and N be (left or right) modules over the same ring, and let f : M → N be a module homomorphism. If M is simple, then f is either the zero homomorphism or injective because the kernel
of f is a submodule of M. If N is simple, then f is either the zero homomorphism or surjective because the image
of f is a submodule of N. If M = N, then f is an endomorphism
of M, and if M is simple, then the prior two statements imply that f is either the zero homomorphism or an isomorphism. Consequently the endomorphism ring
of any simple module is a division ring
. This result is known as Schur's lemma
.
The converse of Schur's lemma is not true in general. For example, the Z-module Q
is not simple, but its endomorphism ring is isomorphic to the field Q.
A common approach to proving a fact about M is to show that the fact is true for the center term of a short exact sequence when it is true for the left and right terms, then to prove the fact for N and M/N. If N has a non-zero proper submodule, then this process can be repeated. This produces a chain of submodules
In order to prove the fact this way, one needs conditions on this sequence and on the modules Mi/Mi + 1. One particularly useful condition is that the length
of the sequence is finite and each quotient module Mi/Mi + 1 is simple. In this case the sequence is called a composition series for M. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module. For example, the Fitting lemma
shows that the endomorphism ring
of a finite length indecomposable module
is a local ring
, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.
The Jordan–Hölder theorem and the Schreier refinement theorem
describe the relationships amongst all composition series of a single module. The Grothendieck group
ignores the order in a composition series and views every finite length module as a formal sum of simple modules. Over semisimple rings, this is no loss as every module is a semisimple module
and so a direct sum
of simple modules. Ordinary character theory provides better arithmetic control, and uses simple CG modules to understand the structure of finite group
s G. Modular representation theory
uses Brauer characters to view modules as formal sums of simple modules, but is also interested in how those simple modules are joined together within composition series. This is formalized by studying the Ext functor
and describing the module category in various ways including quivers
(whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2) and Auslander–Reiten theory
where the associated graph has a vertex for every indecomposable module.
. The Jacobson density theorem states:
In particular, any primitive ring
may be viewed as (that is, isomorphic to) a ring of D-linear operators on some D-space.
A consequence of the Jacobson density theorem is Wedderburn's theorem; namely that any right artinian
simple ring
is isomorphic to a full matrix ring of n by n matrices over a division ring
for some n. This can also be established as a corollary of the Artin–Wedderburn theorem
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically in ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, the simple modules over a ring R are the (left or right) module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
s over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
every cyclic submodule
Cyclic module
In mathematics, more specifically in ring theory, a cyclic module is a module over a ring which is generated by one element. The term is by analogy with cyclic groups, that is groups which are generated by one element.- Definition :...
generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length
Length of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
, and they are analogous to the simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
s in group theory.
In this article, all modules will be assumed to be right unital modules over a ring R.
Examples
ZInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups. These are the cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
s of prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
.
If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal. Conversely, if I is not minimal, then there is a non-zero right ideal J properly contained in I. J is a right submodule of I, so I is not simple.
If I is a right ideal of R, then R/I is simple if and only I is a maximal right ideal: If M is a non-zero proper submodule of R/I, then the preimage of M under the quotient map is a right ideal which is not equal to R and which properly contains I. Therefore I is not maximal. Conversely, if I is not maximal, then there is a right ideal J properly containing I. The quotient map has a non-zero kernel which is not equal to , and therefore is not simple.
Every simple R-module is isomorphic to a quotient R/m where m is a maximal right ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
of R. By the above paragraph, any quotient R/m is a simple module. Conversely, suppose that M is a simple R-module. Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x. The statement that xR = M is equivalent to the surjectivity of the homomorphism that sends r to xr. The kernel of this homomorphism is a right ideal I of R, and a standard theorem states that M is isomorphic to R/I. By the above paragraph, we find that I is a maximal right ideal. Therefore M is isomorphic to a quotient of R by a maximal right ideal.
If k is a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
and G is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, then a group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of G is a left module over the group ring
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
kG. The simple kG modules are also known as irreducible representations. A major aim of representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
is to understand the irreducible representations of groups.
Basic properties of simple modules
The simple modules are precisely the modules of lengthLength of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
1; this is a reformulation of the definition.
Every simple module is indecomposable
Indecomposable module
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 is a weaker notion than simple module:simple means "no proper submodule" N...
, but the converse is in general not true.
Every simple module is cyclic
Cyclic module
In mathematics, more specifically in ring theory, a cyclic module is a module over a ring which is generated by one element. The term is by analogy with cyclic groups, that is groups which are generated by one element.- Definition :...
, that is it is generated by one element.
Not every module has a simple submodule; consider for instance the Z-module Z in light of the first example above.
Let M and N be (left or right) modules over the same ring, and let f : M → N be a module homomorphism. If M is simple, then f is either the zero homomorphism or injective because the kernel
Kernel (algebra)
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. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of f is a submodule of M. If N is simple, then f is either the zero homomorphism or surjective because the image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
of f is a submodule of N. If M = N, then f is an endomorphism
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
of M, and if M is simple, then the prior two statements imply that f is either the zero homomorphism or an isomorphism. Consequently the endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
of any simple module is a division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
. This result is known as Schur's lemma
Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...
.
The converse of Schur's lemma is not true in general. For example, the Z-module Q
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
is not simple, but its endomorphism ring is isomorphic to the field Q.
Simple modules and composition series
If M is a module which has a non-zero proper submodule N, then there is a short exact sequenceA common approach to proving a fact about M is to show that the fact is true for the center term of a short exact sequence when it is true for the left and right terms, then to prove the fact for N and M/N. If N has a non-zero proper submodule, then this process can be repeated. This produces a chain of submodules
In order to prove the fact this way, one needs conditions on this sequence and on the modules Mi/Mi + 1. One particularly useful condition is that the length
Length of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
of the sequence is finite and each quotient module Mi/Mi + 1 is simple. In this case the sequence is called a composition series for M. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module. For example, the Fitting lemma
Fitting lemma
The Fitting lemma, named after the mathematician Hans Fitting, is a basic statement in abstract algebra. Suppose M is a module over some ring...
shows that the endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
of a finite length indecomposable module
Indecomposable module
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 is a weaker notion than simple module:simple means "no proper submodule" N...
is a local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.
The Jordan–Hölder theorem and the Schreier refinement theorem
Schreier refinement theorem
In mathematics, the Schreier refinement theorem of group theory states that any two normal series of subgroups of a given group have equivalent refinements....
describe the relationships amongst all composition series of a single module. The Grothendieck group
Grothendieck group
In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way...
ignores the order in a composition series and views every finite length module as a formal sum of simple modules. Over semisimple rings, this is no loss as every module is a semisimple module
Semisimple module
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring which is a semisimple module over itself is known as an artinian semisimple ring...
and so a direct sum
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
of simple modules. Ordinary character theory provides better arithmetic control, and uses simple CG modules to understand the structure of finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s G. Modular representation theory
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
uses Brauer characters to view modules as formal sums of simple modules, but is also interested in how those simple modules are joined together within composition series. This is formalized by studying the Ext functor
Ext functor
In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...
and describing the module category in various ways including quivers
Quiver (mathematics)
In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation, V, of a quiver assigns a vector space V to each vertex x of the quiver and a linear map V to each...
(whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2) and Auslander–Reiten theory
Auslander–Reiten theory
In algebra, Auslander–Reiten theory studies the representation theory of Artinian rings using techniques such as Auslander–Reiten sequences and Auslander–Reiten quivers. Auslander–Reiten theory was introduced by and developed by them in several subsequent papers.For survey articles on...
where the associated graph has a vertex for every indecomposable module.
The Jacobson density theorem
An important advance in the theory of simple modules was the Jacobson density theoremJacobson density theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R....
. The Jacobson density theorem states:
- Let U be a simple right R-module and write D = EndR(U). Let A be any D-linear operator on U and let X be a finite D-linearly independent subset of U. Then there exists an element r of R such that x·A = x·r for all x in X.
In particular, any primitive ring
Primitive ring
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.- Definition :...
may be viewed as (that is, isomorphic to) a ring of D-linear operators on some D-space.
A consequence of the Jacobson density theorem is Wedderburn's theorem; namely that any right artinian
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
simple ring
Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
is isomorphic to a full matrix ring of n by n matrices over a division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
for some n. This can also be established as a corollary of the Artin–Wedderburn theorem
Artin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
.
See also
- Semisimple moduleSemisimple moduleIn mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring which is a semisimple module over itself is known as an artinian semisimple ring...
s are modules that can be written as a sum of simple submodules - Simple groupSimple groupIn mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
s are similarly defined to simple modules - Irreducible idealIrreducible idealIn mathematics, an ideal of a commutative ring is said to be irreducible if it cannot be written as a finite intersection of ideals properly containing it....
.