Indecomposable module
Encyclopedia
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"
,
while indecomposable "not expressible as
".
A direct sum of indecomposables is called completely decomposable; this is weaker than being semisimple
, which is a direct sum of simple module
s.
field
or PID
,
and underlies Jordan normal form
of operators.
s are vector space
s. A vector space is indecomposable if and only if its dimension is 1. So every vector space is completely decomposable (indeed, semisimple), with infinitely many summands if the dimension is infinite.
are classified by the
structure theorem for finitely generated modules over a principal ideal domain
:
the primary decomposition is a decomposition into indecomposable modules,
so every finitely-generated module over a PID is completely decomposable.
Explicitly, the modules of the form
for prime ideal
s p (including p=0, which yields R) are indecomposable. Every finitely-generated R-module is a direct sum of these. Note that this is simple if and only if n=1 (or p=0); for example, the cyclic group of order 4, Z/4, is indecomposable but not simple – it has the subgroup 2Z/4 of order 2, but this does not have a complement.
Over the integer
s Z, modules are abelian group
s. A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form
for some prime number
p and some positive integer n. Every finitely-generated abelian group is a direct sum of (finitely many) indecomposable abelian groups.
There are, however, other indecomposable abelian groups which are not finitely generated; the rational number
s Q form the simplest example.
For a fixed positive integer n, consider the ring R of n-by-n matrices
with entries from the real number
s (or from any other field K). Then Kn is a left R-module (the scalar multiplication is matrix multiplication
). This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of (finitely or infinitely many) copies of this module Kn.
is indecomposable. The converse is not true in general, as is shown by the second example above.
By looking at the endomorphism ring
of a module, one can tell whether the module is indecomposable: if and only if the endomorphism ring does not contain an idempotent different from 0 and 1. (If f is such an idempotent endomorphism of M, then M is the direct sum of ker(f) and im(f).)
A module of finite length
is indecomposable if and only if its endomorphism ring is local
. Still more information about endomorphisms of finite-length indecomposables is provided by the Fitting lemma
.
In the finite-length situation, decomposition into indecomposables is particularly useful, because of the Krull-Schmidt theorem: every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique (meaning that if you have a different decomposition into indecomposable, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic).
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a 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...
is indecomposable if it is non-zero and cannot be written as 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 two non-zero submodules.
Indecomposable is a weaker notion than simple module
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules 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 means "no proper submodule"
,while indecomposable "not expressible as
".A direct sum of indecomposables is called completely decomposable; this is weaker than being semisimple
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...
, which is a direct sum of simple module
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules 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...
s.
Motivation
In many situations, all modules of interest are completely decomposable; the indecomposable modules can then be thought of as the "basic building blocks", the only objects that need to be studied. This is the case for modules over afield
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...
or PID
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
,
and underlies Jordan normal form
Jordan normal form
In linear algebra, a Jordan normal form of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some basis...
of operators.
Field
Modules over fieldField (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...
s are vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s. A vector space is indecomposable if and only if its dimension is 1. So every vector space is completely decomposable (indeed, semisimple), with infinitely many summands if the dimension is infinite.
PID
Finitely-generated modules over PIDsPrincipal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
are classified by the
structure theorem for finitely generated modules over a principal ideal domain
Structure theorem for finitely generated modules over a principal ideal domain
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in...
:
the primary decomposition is a decomposition into indecomposable modules,
so every finitely-generated module over a PID is completely decomposable.
Explicitly, the modules of the form
for prime idealPrime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s p (including p=0, which yields R) are indecomposable. Every finitely-generated R-module is a direct sum of these. Note that this is simple if and only if n=1 (or p=0); for example, the cyclic group of order 4, Z/4, is indecomposable but not simple – it has the subgroup 2Z/4 of order 2, but this does not have a complement.
Over the integer
Integer
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...
s Z, modules are abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s. A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form
for some prime numberPrime 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...
p and some positive integer n. Every finitely-generated abelian group is a direct sum of (finitely many) indecomposable abelian groups.
There are, however, other indecomposable abelian groups which are not finitely generated; the rational number
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...
s Q form the simplest example.
For a fixed positive integer n, consider the ring R of n-by-n matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
with entries from the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s (or from any other field K). Then Kn is a left R-module (the scalar multiplication is matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
). This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of (finitely or infinitely many) copies of this module Kn.
Facts
Every simple moduleSimple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules 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...
is indecomposable. The converse is not true in general, as is shown by the second example above.
By looking at 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 module, one can tell whether the module is indecomposable: if and only if the endomorphism ring does not contain an idempotent different from 0 and 1. (If f is such an idempotent endomorphism of M, then M is the direct sum of ker(f) and im(f).)
A module 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...
is indecomposable if and only if its endomorphism ring is local
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...
. Still more information about endomorphisms of finite-length indecomposables is provided by 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...
.
In the finite-length situation, decomposition into indecomposables is particularly useful, because of the Krull-Schmidt theorem: every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique (meaning that if you have a different decomposition into indecomposable, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic).