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.
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 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.
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.
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).
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
Every finitely generated module M over a principal ideal domain R is isomorphic to one of the form where
are called the elementary divisors of M. In a PID, nonzero primary ideals are powers of primes, and so
Since PID's are Noetherian rings, this can be seen as a manifestation of the Lasker-Noether theorem.
One proof proceeds as follows: This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors.
for a positive integer n. Since every free module is projective module, then exists right inverse of the projection map (it suffices to lift each of the generators of M/tM into M).
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.
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
, 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
, this yields various canonical forms: 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.
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.
The primary decomposition generalizes to finitely generated modules over commutative Noetherian rings, and this result is called the Lasker–Noether theorem.
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.
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. However, over a Dedekind domain the ideal class group is the only obstruction, and the structure theorem generalizes to finitely generated modules over a Dedekind domain with minor modifications.
There is still a unique torsion part, with a torsionfree complement (unique up to isomorphism), but a torsionfree module over a Dedekind domain is no longer necessarily free.
Torsionfree modules over a Dedekind domain are determined (up to isomorphism) by rank and Steinitz class (which takes value in the ideal class group), and the decomposition into a direct sum of copies of R (rank one free modules) is replaced by a direct sum into rank one projective modules: the individual summands are not uniquely determined, but the Steinitz class (of the sum) is.
Similarly for modules that are not finitely generated, one cannot expect such a nice decomposition: even the number of factors may vary.
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.