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:

${\displaystyle M\cong \bigoplus _{i}R/(d_{i})=R/(d_{1})\oplus R/(d_{2})\oplus \cdots \oplus R/(d_{n}).}$

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.

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.

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

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.