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 over a principal ideal domain (PID) 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.

Contents 1 Statement 1.1 Invariant factor decomposition 1.2 Primary decomposition 2 Proofs 3 Corollaries 4 Uniqueness 5 Generalizations 5.1 Groups 5.2 Primary decomposition 5.3 Indecomposable modules 5.4 Non-finitely generated modules 6 References 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, since a basis for a finitely generated module over R might not exist. 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 of 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 usually appears in the following two forms.

Invariant factor decomposition For every finitely generated module M over a principal ideal domain R, there is a unique decreasing sequence of proper ideals {displaystyle (d_{1})supseteq (d_{2})supseteq cdots supseteq (d_{n})} such that M is isomorphic to the sum of cyclic modules: {displaystyle Mcong bigoplus _{i}R/(d_{i})=R/(d_{1})oplus R/(d_{2})oplus cdots oplus R/(d_{n}).} The generators {displaystyle d_{i}} of the ideals are unique up to multiplication by a unit, and are called invariant factors 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 {displaystyle d_{1},|,d_{2},|,cdots ,|,d_{n}} . The free part is visible in the part of the decomposition corresponding to factors {displaystyle d_{i}=0} . Such factors, if any, occur at the end of the sequence.

While the direct sum is uniquely determined by M, the isomorphism giving the decomposition itself is not unique in general. For instance if R 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 dim M > 1).

The nonzero {displaystyle d_{i}} elements, together with the number of {displaystyle d_{i}} which are zero, form a complete set of invariants 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: {displaystyle R^{f}oplus bigoplus _{i}R/(d_{i})=R^{f}oplus R/(d_{1})oplus R/(d_{2})oplus cdots oplus R/(d_{n-f})} where the visible {displaystyle d_{i}} are nonzero, and f is the number of {displaystyle d_{i}} '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 {displaystyle bigoplus _{i}R/(q_{i})} where {displaystyle (q_{i})neq R} and the {displaystyle (q_{i})} are primary ideals. The {displaystyle q_{i}} are unique (up to multiplication by units).

The elements {displaystyle q_{i}} are called the elementary divisors of M. In a PID, nonzero primary ideals are powers of primes, and so {displaystyle (q_{i})=(p_{i}^{r_{i}})=(p_{i})^{r_{i}}} . When {displaystyle q_{i}=0} , the resulting indecomposable module is {displaystyle R} itself, and this is inside the part of M that is a free module.

The summands {displaystyle 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 a completely decomposable module. Since PID's are Noetherian rings, this can be seen as a manifestation of the Lasker-Noether theorem.

As before, it is possible to write the free part (where {displaystyle q_{i}=0} ) separately and express M as: {displaystyle R^{f}oplus (bigoplus _{i}R/(q_{i}))} where the visible {displaystyle q_{i}} 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 {displaystyle 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, so it is isomorphic to {displaystyle R^{n}} 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 {displaystyle M=tMoplus F} . For a prime element p in R we can then speak of {displaystyle N_{p}={min tMmid exists i,mp^{i}=0}} . This is a submodule of tM, and it turns out that each Np is a direct sum of cyclic modules, and that tM is a direct sum of Np 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 {displaystyle R=K} . Since fields have no non-trivial ideals, every finitely generated vector space is free.

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

Let T be a linear operator on a finite-dimensional vector space V over K. Taking {displaystyle R=K[T]} , the algebra of polynomials with coefficients in K evaluated at T, yields structure information about T. V can be viewed as a finitely generated module over {displaystyle K[T]} . 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 {displaystyle 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: {displaystyle 0

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