In
mathematics, in the field of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the
fundamental theorem of finitely generated abelian groups
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
and roughly states that
finitely generated modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
over a
principal ideal domain
In mathematics, 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 princi ...
(PID) can be uniquely decomposed in much the same way that
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s have a
prime factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are su ...
. The result provides a simple framework to understand various canonical form results for
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are ofte ...
over
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
.
Statement
When a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
over a field ''F'' has a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb
Traditionally, a finite verb (from la, fīnītus, past particip ...
generating set, then one may extract from it a
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
consisting of a finite number ''n'' of vectors, and the space is therefore
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to ''F''
''n''. The corresponding statement with the ''F'' generalized to a
principal ideal domain
In mathematics, 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 princi ...
''R'' is no longer true, since a basis for a
finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type.
Related concepts in ...
over ''R'' might not exist. However such a module is still isomorphic to a
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of some module ''R
n'' with ''n'' finite (to see this it suffices to construct the morphism that sends the elements of the canonical basis of ''R
n'' to the generators of the module, and take the quotient by its
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...
.) By changing the choice of generating set, one can in fact describe the module as the quotient of some ''R
n'' by a particularly simple
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
, and this is the structure theorem.
The structure theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms.
Invariant factor decomposition
For every finitely generated module over a principal ideal domain , there is a unique decreasing sequence of
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map fo ...
ideals such that is isomorphic to the
sum of
cyclic module In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z- ...
s:
:
The generators
of the ideals are unique up to multiplication by a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
, and are called
invariant factor The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain.
If R is a PID and M a finitely generated R-module, then
:M\cong R^r\o ...
s of ''M''. Since the ideals should be proper, these factors must not themselves be invertible (this avoids trivial factors in the sum), and the inclusion of the ideals means one has divisibility
. The free part is visible in the part of the decomposition corresponding to factors
. Such factors, if any, occur at the end of the sequence.
While the direct sum is uniquely determined by , the isomorphism giving the decomposition itself is ''not unique'' in general. For instance if is actually a field, then all occurring ideals must be zero, and one obtains the decomposition of a finite dimensional vector space into a direct sum of one-dimensional
subspaces; the number of such factors is fixed, namely the dimension of the space, but there is a lot of freedom for choosing the subspaces themselves (if ).
The nonzero
elements, together with the number of
which are zero, form a
complete set of invariants In mathematics, a complete set of invariants for a classification problem is a collection of maps
:f_i : X \to Y_i
(where X is the collection of objects being classified, up to some equivalence relation \sim, and the Y_i are some sets), such that ...
for the module. Explicitly, this means that any two modules sharing the same set of invariants are necessarily isomorphic.
Some prefer to write the free part of ''M'' separately:
:
where the visible
are nonzero, and ''f'' is the number of
's in the original sequence which are 0.
Primary decomposition
:Every finitely generated module ''M'' over a principal ideal domain ''R'' is isomorphic to one of the form
::
:where
and the
are
primary ideal In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y'n'' is also an element of ''Q'', for some ''n'' > 0. ...
s. The
are unique (up to multiplication by units).
The elements
are called the ''elementary divisors'' of ''M''. In a PID, nonzero primary ideals are powers of primes, and so
. When
, the resulting indecomposable module is
itself, and this is inside the part of ''M'' that is a free module.
The summands
are
indecomposable, so the primary decomposition is a decomposition into indecomposable modules, and thus every finitely generated module over a PID is a
completely decomposable module. Since PID's are
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noeth ...
s, this can be seen as a manifestation of the
Lasker-Noether theorem
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are related ...
.
As before, it is possible to write the free part (where
) separately and express ''M'' as:
:
where the visible
are nonzero.
Proofs
One proof proceeds as follows:
* Every finitely generated module over a PID is also
finitely presented because a PID is Noetherian, an even stronger condition than
coherence.
* Take a presentation, which is a map
(relations to generators), and put it in
Smith normal form In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can b ...
.
This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors.
Another outline of a proof:
* Denote by ''tM'' the
torsion submodule of ''M''. Then ''M''/''tM'' is a finitely generated
torsion free module, and such a module over a commutative PID is a
free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field i ...
of finite
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
, so it is isomorphic to
for a positive integer ''n''. This free module can be
embedded as a submodule ''F'' of ''M'', such that the embedding splits (is a right inverse of) the projection map; it suffices to lift each of the generators of ''F'' into ''M''. As a consequence
.
* For a
prime element
In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish pri ...
''p'' in ''R'' we can then speak of
. This is a submodule of ''tM'', and it turns out that each ''N''
''p'' is a direct sum of cyclic modules, and that ''tM'' is a direct sum of ''N''
''p'' for a finite number of distinct primes ''p''.
* Putting the previous two steps together, ''M'' is decomposed into cyclic modules of the indicated types.
Corollaries
This includes the classification of finite-dimensional vector spaces as a special case, where
. Since fields have no non-trivial ideals, every finitely generated vector space is free.
Taking
yields the
fundamental theorem of finitely generated abelian groups
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
.
Let ''T'' be a linear operator on a finite-dimensional vector space ''V'' over ''K''. Taking