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 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.

Statement

When a vector space over a field F has a finite generating set, then one may extract from it a basis consisting of a finite number n of vectors, and the space is therefore isomorphic to Fⁿ. The corresponding statement with the F generalized to a principal ideal domain R is no longer true, as a finitely generated module over R need not have any basis. However such a module is still isomorphic to a quotient of some module Rⁿ with n finite (to see this it suffices to construct the morphism that sends the elements of the canonical basis Rⁿ to the generators of the module, and take the quotient by its kernel. By changing the choice of generating set, one can in fact describe the module as the quotient of some Rⁿ by a particularly simple submodule, and this is the structure theorem.

The structure theorem for finitely generated modules over a principal ideal domain has two statements, which are equivalent by the Chinese remainder theorem:

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.

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.

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 $R^r \to R^g$ (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.

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 of finite rank. As a result, $M= tM\oplus F$ where $F\cong R^n$ for a positive integer n.
• For a prime p in R we can then speak of $N_p= \{m\in tM\mid \exists i, mp^i=0\}$ for each prime p. 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 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 – an operator on a vector space is the same as an algebra representation of the polynomial algebra in one variable – where the last invariant factor is the minimal polynomial, and the product of invariant factors 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)

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).

Generalizations

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.

Primary decomposition

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

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.

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.

Another issue that arises with non-finitely generated modules is that there are torsion-free modules which are not free. For instance, consider the ring Z of integers. A classical example of a torsion-free module which is not free is the Baer–Specker group, the group of all sequences of integers under termwise addition. In general, the question of which infinitely generated torsion-free abelian groups are free depends on which large cardinals exist. A consequence is that any structure theorem for infinitely generated modules depends on a choice of set theory axioms and may be invalid under a different choice.

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, ISBN 978-0-471-43334-7, MR2286236 
• 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 

• Jacobson, Nathan (1985), Basic algebra. I (2 ed.), New York: W. H. Freeman and Company, pp. xviii+499, ISBN 0-7167-1480-9, MR780184 

• Lam, T. Y. (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Springer-Verlag, ISBN 978-0-387-98428-5