Take a presentation, which is a map (relations to generators), and put it in .
Smith normal form
One proof proceeds as follows:
This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors.
Another outline of a proof:
Where the map is a projection. 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: 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). By splitting lemma (left split) M splits into: .
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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.
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 . 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 , this yields various canonical forms:
![{\displaystyle R=K[T]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57eb6a1c52311fc6ba483dcf7b0b1fa3a64c06d5)
![{\displaystyle K[T]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7bc4920fa11e40cf90259a765847f0b9e33a635)
![{\displaystyle K[T]/p(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98731e0c35ee66043737bd930f744078e7e326b7)
Generalizations[edit]
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Groups[edit]
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[edit]
The primary decomposition generalizes to finitely generated modules over commutative Noetherian rings, and this result is called the Lasker–Noether theorem.
Indecomposable modules[edit]
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.
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.
Non-finitely generated modules[edit]
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 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. Then Q is a torsion-free Z-module which is not free. Another classical example of such a module 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.
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