Structure theorem for finitely generated modules over a principal ideal domain

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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 much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.

The name, which does not cite the discoverer, is an example of obliteration by incorporation.

[edit] Statement

The structure theorem for finitely generated modules over a principal ideal domain has two statements, which are equivalent by the Chinese remainder theorem^{[citation needed]}:

[edit] Invariant factor decomposition

Every finitely generated module M over a principal ideal domain R is isomorphic to a unique one of the form

$\oplus_i R/(d_i) \quad\equiv\quad R/(d_1)\oplus R/(d_2)\oplus\cdots\oplus R/(d_n)$

where $d_i \neq 1$ and $d_i \vert d_{i+1}$ .

The elements

d i

(up to unit) are a complete set of invariants for finitely generated R-modules, and are called invariant factors.

The ideals

(d i)

are unique; the elements

d i

are unique up to multiplication by a unit, but the order is unique.

Some prefer to separate out the free part and write M as:

$R^f \oplus_i R/(d_i) \quad\equiv\quad R^f \oplus R/(d_1)\oplus R/(d_2)\oplus\cdots\oplus R/(d_{n-f})$

where $d_i \neq 0, 1$ and $d_i \vert d_{i+1}$ . The free part is where (in the first formulation) $d i = 0$ ; these occur at the end, as everything divides zero.

[edit] Primary decomposition

Every finitely generated module M over a principal ideal domain R is isomorphic to a unique one of the form

$\oplus_i R/(q_i)$

where $q_i \neq 1$ and the

(q i)

are primary ideals. The ideals

(q i)

are unique (up to order); the elements

q i

are unique up to multiplication by a unit, and are called the elementary divisors.

Note that in a PID, primary ideals are powers of primes, so $(q_i)=(p_i^{r_i}) = (p_i)^{r_i}$ .

The summands $R / (q i)$ are indecomposable, so the primary decomposition is a decomposition into indecomposable modules, and thus every finitely generated module over a PID is completely decomposable.

Some prefer to separate out the free part (where $q i = 0$ ) and write M as:

$R^f \oplus_i R/(q_i)$

where $q_i \neq 0, 1$ and the $(q i)$ are primary ideals.

[edit] Proofs

One proof proceeds as follows:

Every finitely generated module over a PID is finitely presented
Take a presentation, which is a map $R^r \to R^g$ (relations to generators), and put it in Smith normal form

This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors.

An alternative proof using some more abstract (and useful) machinery proceeds as follows:

A finitely generated module is projective if and only if it is locally free.
PIDs are [Dedekind domains], i.e., they are noetherian and their localizations are discrete valuation rings (DVRs).
Torsion free modules over DVRs are free. Hence, torsion free modules over PIDs are projective.
M/tM is torsion free, hence projective. Thus, M can be written as a direct sum of its torsion free and free parts.
If M is projective, so is a direct summand of a free module F = M + N. One proves that N is locally 0, and hence is 0. Therefore, M is free.
If M is torsion, it is the quotient of a free module. Using the ideas of the previous part, one proves it is a quotient by a free submodule, which must have rank equal to the original module. That is, torsion modules are finitely presented. Now use Smith normal form.

[edit] Corollaries

This includes the classification of finite-dimensional vector spaces as a special case, where $R = K$ . Since fields have no non-trivial ideals, every finitely generated vector space is free.

Taking $R=\mathbb{Z}$ yields the fundamental theorem of finitely generated abelian groups.

Taking $R = K [T]$ classifies linear operators on a finite-dimensional vector space, where the first invariant factor is the minimal polynomial, and the last invariant factor is the characteristic polynomial. Combined with a standard matrix form for $K [T] / p (T)$ , this yields various canonical forms:

invariant factors + companion matrix yields Frobenius normal form (aka, rational canonical form)
primary decomposition + companion matrix yields primary rational canonical form
primary decomposition + Jordan blocks yields Jordan canonical form (this latter only holds over an algebraically closed field)

[edit] Uniqueness

While the invariants (rank, invariant factors, and elementary divisors) are unique, the isomorphism between M and its canonical form is not unique, and does not even preserve the direct sum decomposition. This follows because there are non-trivial automorphisms of these modules which do not preserve the summands.

However, one has a canonical torsion submodule T, and similar canonical submodules corresponding to each (distinct) invariant factor, which yield a canonical sequence:

$0 < \cdots < T < M.$

Compare composition series in Jordan-Hölder theorem.

For instance, if $M \approx \mathbf{Z} \oplus \mathbf{Z}/2$ , and $(1,0),(0,1)$ is one basis, then $(1,1),(0,1)$ is another basis, and the change of basis matrix $\begin{bmatrix}1 & 1 \\0 & 1\end{bmatrix}$ does not preserve the summand $\mathbf{Z}$ . However, it does preserve the $\mathbf{Z}/2$ summand, as this is the torsion submodule (equivalently here, the 2-torsion elements).

[edit] Generalizations

[edit] Groups

The Jordan-Hölder theorem is a more general result for finite groups (or modules over an arbitrary ring). In this generality, one obtains a composition series, rather than a direct sum.

The Krull-Schmidt theorem and related results give conditions under which a module has something like a primary decomposition, a decomposition as a direct sum of indecomposable modules in which the summands are unique up to order.

[edit] Primary decomposition

The primary decomposition generalizes to finitely generated modules over commutative Noetherian rings, and this result is called the Lasker–Noether theorem.

[edit] Indecomposable modules

By contrast, unique decomposition into indecomposable submodules does not generalize as far, and the failure is measured by the ideal class group, which vanishes for PIDs.

For rings that are not principal ideal domains, unique decomposition need not even hold for modules over a ring generated by two elements. For the ring R=Z[√−5], both the module R and its submodule M generated by 2 and 1+√−5 are indecomposable. While R is not isomorphic to M, R ⊕ R is isomorphic to M ⊕ M; thus the images of the M summands give indecomposable submodules L₁, L₂ < R ⊕ R which give a different decomposition of R ⊕ R. The failure of uniquely factorizing R ⊕ R into a direct sum of indecomposable modules is directly related (via the ideal class group) to the failure of the unique factorization of elements of R into irreducible elements of R.

[edit] Non-finitely generated modules

Similarly for modules that are not finitely generated, one cannot expect such a nice decomposition: even the number of factors may vary. There are Z-submodules A of Q⁴ which are simultaneously direct sums of two indecomposable modules and direct sums of three indecomposable modules, showing the analogue of the primary decomposition cannot hold for infinitely generated modules, even over the integers, Z.

[edit] References

Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: Wiley, MR 2286236, ISBN 978-0-471-43334-7
Hungerford, Thomas W. (1980), Algebra, New York: Springer, pp. 218–226, Section IV.6: Modules over a Principal Ideal Domain, ISBN 978-0-387-90518-1
Lam, T. Y. (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Springer-Verlag, ISBN 978-0-387-98428-5

Structure theorem for finitely generated modules over a principal ideal domain

From Wikipedia, the free encyclopedia

Contents

[edit] Statement

[edit] Invariant factor decomposition

[edit] Primary decomposition

[edit] Proofs

[edit] Corollaries

[edit] Uniqueness

[edit] Generalizations

[edit] Groups

[edit] Primary decomposition

[edit] Indecomposable modules

[edit] Non-finitely generated modules

[edit] References

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