The Structure Theorem for Modules over a PID

A module over a general ring can be arbitrarily complicated: it need not have a basis, a submodule of a free module need not be free, and there is no classification. Restrict the ring to a principal ideal domain and restrict the module to finitely generated, and the picture collapses to a single statement. The module is a free part $R^{r}$ plus a finite direct sum of cyclic pieces $R / (a)$ , and the ideals $(a)$ that appear are determined by the module up to reordering.

This is the structure theorem for finitely generated modules over a PID. It is the source of two things at once: the classification of finite abelian groups (take $R = Z$ ) and the canonical forms of a linear operator (take $R = F [x]$ ).

Rank, torsion, and the annihilator

Throughout, $R$ is a PID and $M$ is an $R$ -module. Three invariants organize the whole theory.

Over a field this recovers the dimension of a vector space, and a maximal independent set is a basis. Over a general PID the rank still counts independence, but a rank- $n$ module need not have a basis of $n$ elements: an element killed by a nonzero ring element can never sit in an independent set, so $Z \oplus Z /2 Z$ has rank $1$ yet is not free.

The last containment is the module version of a divisibility statement that will reappear as the invariant-factor chain. When $R = Z$ it is Lagrange's theorem: the order of an element divides the order of the group.

A PID is Noetherian — every ascending chain of ideals stabilizes — because any union of a chain of ideals is again a principal ideal, generated by an element that already lives in one link of the chain. Consequently every submodule of a finitely generated module over a PID is again finitely generated.¹

The stacked-basis theorem

The whole chapter rests on how a submodule sits inside a free module. Over a field, a subspace of $F^{n}$ is again $F^{m}$ and any basis extends; over a PID the analogue holds, but the two bases are linked by scalars.

A single basis of $M$ serves for both modules: multiply its first $m$ vectors by the scalars $a_{i}$ to obtain a basis of $N$ .

The stacked-basis theorem: one basis of

M

works for both modules, its first

m

vectors scaled by

a_{1} ∣ a_{2} ∣ \dots ∣ a_{m}

to give a basis of the submodule

N

Existence: invariant factors

Present $M$ by generators and relations. If $M$ is generated by $n$ elements, there is a surjection $π : R^{n} \to M$ , and by the first isomorphism theorem $M ≅ R^{n} / ker π$ . The kernel is a submodule of the free module $R^{n}$ , so the stacked-basis theorem applies to it. Quotienting the free module by its scaled sub-basis leaves exactly a direct sum of cyclic modules.

The invariant-factor decomposition: a free part of rank

r

and torsion cyclic summands whose annihilators form the divisibility chain.

The divisibility chain is what pins the decomposition down. Without it the ideals $(a_{i})$ are not unique: $Z /6 ≅ Z /2 \oplus Z /3$ , so the same module can be written with different lists of cyclic factors. The condition $a_{1} ∣ a_{2} ∣ \dots ∣ a_{m}$ singles out one list.

Computing the decomposition from a relations matrix

The decomposition is computable. When $R$ is a Euclidean domain (so $R = Z$ or $R = F [x]$ in the applications), write the generators $y_{1}, \dots, y_{m}$ of $ker π$ in the basis $x_{1}, \dots, x_{n}$ of $R^{n}$ ; the coefficients form the $m \times n$ relations matrix $A = (a_{ij})$ . Elementary row operations change the choice of generators for $ker π$ , elementary column operations change the basis of $R^{n}$ , and neither touches the module $M$ . Diagonalizing $A$ by these operations therefore reads off the invariant factors.³

Worked example. Let $M = Z^{3} / K$ , where $K$ is generated by the three relations

2 x_{1} + 4 x_{2} + 6 x_{3} 4 x_{1} + 7 x_{2} + 8 x_{3} 6 x_{1} + 8 x_{2} + 14 x_{3} = 0, = 0, = 0, A = 2464786814 .

No entry is a unit, so take the smallest entry $2$ as pivot and clear its column and row:

A R_{2} \to R_{2} - 2 R_{1} 206 4 - 1 8 6 - 4 14 R_{3} \to R_{3} - 3 R_{1} 200 4 - 1 - 4 6 - 4 - 4

C_{2} \to C_{2} - 2 C_{1} 200 0 - 1 - 4 6 - 4 - 4 C_{3} \to C_{3} - 3 C_{1} 200 0 - 1 - 4 0 - 4 - 4 .

The pivot $2$ does not divide the interior entry $- 1$ , so the divisibility chain would fail; repair it by moving $- 1$ into the pivot position. Add row $2$ to row $1$ , swap columns $1$ and $2$ , and normalize the sign:

R_{1} \to R_{1} + R_{2} 200 - 1 - 1 - 4 - 4 - 4 - 4 C_{1} \leftrightarrow C_{2} - 1 - 1 - 4 200 - 4 - 4 - 4 R_{1} \to - R_{1} 1 - 1 - 4 - 2 00 4 - 4 - 4

and clear again with the new pivot $1$ :

R_{2} \to R_{2} + R_{1} R_{3} \to R_{3} + 4 R_{1} 100 - 2 - 2 - 8 4012 C_{2} \to C_{2} + 2 C_{1} C_{3} \to C_{3} - 4 C_{1} 100 0 - 2 - 8 0012 .

The remaining $2 \times 2$ block has smallest entry $- 2$ , which divides both other entries:

R_{2} \to - R_{2} 100 02 - 8 0012 R_{3} \to R_{3} + 4 R_{2} 1000200012 .

The diagonal is $1, 2, 12$ with $1 ∣ 2 ∣ 12$ . The unit entry $1$ contributes the zero summand $Z / (1) = 0$ , so

M ≅ Z /2 Z \oplus Z /12 Z,

an abelian group of order $24$ with invariant factors $2 ∣ 12$ . Its elementary divisors follow by factoring: $2$ and $12 = 4 \cdot 3$ give $2, 4, 3$ , so also $M ≅ Z /2 \oplus Z /4 \oplus Z /3$ . As a check, the product of the invariant factors is $det A$ up to sign: $det A = - 24$ and $1 \cdot 2 \cdot 12 = 24$ .

Existence: elementary divisors

The invariant factors can be broken further. In a PID every nonzero non-unit factors into prime powers, and the Chinese Remainder Theorem splits a cyclic module along coprime factors: if $a = u p_{1}^{α_{1}} \dots p_{s}^{α_{s}}$ with distinct primes $p_{i}$ , then the ideals $(p_{i}^{α_{i}})$ are pairwise comaximal and

R / (a) ≅ R / (p_{1}^{α_{1}}) \oplus \dots \oplus R / (p_{s}^{α_{s}}) .

Applying this to each cyclic factor in the invariant-factor form gives the second decomposition.

Grouping the summands by prime gives the primary decomposition: the torsion part is $N_{1} \oplus \dots \oplus N_{k}$ where $N_{i}$ , the $p_{i}$ -primary component, collects every element annihilated by some power of $p_{i}$ . Each $N_{i}$ is itself a direct sum of cyclic modules with annihilators powers of the single prime $p_{i}$ .⁴

The two decompositions are two readings of one grid. Lay the elementary divisors out with one row per prime and one column per invariant factor: the entry in row $p$ , column $a_{j}$ is the $p$ -power dividing $a_{j}$ . Reading down a column multiplies the prime powers into an invariant factor; reading across a row lists the elementary divisors for that prime.

The conversion grid. Each column multiplies to an invariant factor; each row lists the elementary divisors for one prime. Columns are padded with

1

so all rows have equal length.

To go from elementary divisors to invariant factors: sort the powers of each prime into nondecreasing order, pad the shorter lists with $1$ 's so every prime has the same count, and multiply the largest power of each prime for the largest invariant factor, the next largest for the next, and so on.

Algorithm:

\textsc{ToInvariantFactors}

— assemble invariant factors from elementary divisors

1
input: multiset $E$ of prime powers $p^\alpha$
2
for each distinct prime $p$ occurring in $E$ do
3
collect its powers into a list, sorted in nondecreasing exponent
4
let $w \gets$ the maximum list length over all primes
5
pad every prime's list on the left with copies of $1$ up to length $w$
6
for $j \gets w$ down to $1$ do
7
$a_j \gets$ product over all primes of the $j$ -th entry of that prime's list
8
return $a_1 \mid a_2 \mid \cdots \mid a_w$

Worked example. Find the invariant factors of $M = Z /2 \oplus Z /4 \oplus Z /3 \oplus Z /9 \oplus Z /5$ . The elementary divisors are $2, 4, 3, 9, 5$ . Sort by prime and pad to the longest list (length $2$ ):

prime	smaller	larger
$2$	$2$	$4$
$3$	$3$	$9$
$5$	$1$	$5$

Column products give the invariant factors:

a_{2} = 4 \cdot 9 \cdot 5 = 180, a_{1} = 2 \cdot 3 \cdot 1 = 6,

so $M ≅ Z /6 \oplus Z /180$ with $6 ∣ 180$ . As a check, $6 \cdot 180 = 1080 = 2 \cdot 4 \cdot 3 \cdot 9 \cdot 5 = ∣ M ∣$ .

Uniqueness

Existence gives a decomposition; the theorem also asserts that it is essentially unique. The proof reduces to counting dimensions of vector spaces over the residue field $F = R / (p)$ .

The theorem yields a decision procedure for isomorphism: put both modules in invariant factor (or elementary divisor) form and compare the lists.

	Invariant factor form	Elementary divisor form
Summands	$R / (a_{1}), \dots, R / (a_{m})$	$R / (p_{i}^{α_{i}})$ , prime powers
Constraint	$a_{1} ∣ a_{2} ∣ \dots ∣ a_{m}$	each annihilator a prime power
Count	as few cyclic factors as possible	as many, as simple as possible
Reads off	minimal generators; $Ann = (a_{m})$	primary components
Uniqueness	the list $(a_{1}, \dots, a_{m})$	the multiset ${p_{i}^{α_{i}}}$
Bridge	CRT splits each $R / (a_{i})$	pack prime powers back into a chain

The case $R = Z$

Take $R = Z$ . A finitely generated $Z$ -module is exactly a finitely generated abelian group, cyclic modules $Z / (a)$ are cyclic groups, and the two forms of the theorem are the two forms of the classification of finite abelian groups. The free rank $r$ counts the copies of $Z$ ; the torsion part is the finite subgroup.

For example, take the abelian group of order $24$ with invariant factors $2$ and $12$ . The chain $2 ∣ 12$ holds, so this is the invariant-factor form $Z_{2} \oplus Z_{12}$ . Splitting $12 = 4 \cdot 3$ by CRT gives the elementary-divisor form $Z_{2} \oplus Z_{4} \oplus Z_{3}$ , with elementary divisors $2, 4, 3$ — one prime power at a time.

One abelian group of order

24

in both forms: two invariant factors

2 ∣ 12

, or three prime-power elementary divisors, bridged by splitting

Z_{12}

along coprime factors.

Enumerating abelian groups of a given order is now bookkeeping on prime powers. For order $p^{3}$ , the elementary divisors for the single prime $p$ are the partitions of $3$ : ${p^{3}}$ , ${p^{2}, p}$ , ${p, p, p}$ , giving $Z_{p^{3}}$ , $Z_{p^{2}} \oplus Z_{p}$ , and $Z_{p}^{3}$ — three groups, one per partition of the exponent.

Worked example. List all abelian groups of order $72 = 2^{3} \cdot 3^{2}$ . Elementary divisors for $2$ are partitions of $3$ ( ${8}$ , ${4, 2}$ , ${2, 2, 2}$ ); for $3$ , partitions of $2$ ( ${9}$ , ${3, 3}$ ). Every pairing gives one group: $3 \times 2 = 6$ groups. Converting each pairing to invariant factors (pad, take column products):

elementary divisors	invariant factors	group
$8; 9$	$72$	$Z_{72}$
$8; 3, 3$	$3 ∣ 24$	$Z_{3} \oplus Z_{24}$
$4, 2; 9$	$2 ∣ 36$	$Z_{2} \oplus Z_{36}$
$4, 2; 3, 3$	$6 ∣ 12$	$Z_{6} \oplus Z_{12}$
$2, 2, 2; 9$	$2 ∣ 2 ∣ 18$	$Z_{2} \oplus Z_{2} \oplus Z_{18}$
$2, 2, 2; 3, 3$	$2 ∣ 6 ∣ 6$	$Z_{2} \oplus Z_{6} \oplus Z_{6}$

Each invariant-factor list multiplies to $72$ and satisfies the divisibility chain, and no two lists agree, so these six groups are pairwise nonisomorphic.

Two specializations

The statement is neutral about which PID $R$ is, and two choices carry it into linear algebra:

$R = Z$ classifies finite abelian groups, as above.
$R = F [x]$ classifies linear operators. A finite-dimensional vector space $V$ with a chosen operator $T$ becomes an $F [x]$ -module by letting $x$ act as $T$ . Because $dim V < \infty$ , this module is finitely generated and torsion. Its invariant factors are polynomials, and the two canonical decompositions become the rational canonical form and the Jordan canonical form.

The dictionary is exact: the largest invariant factor becomes the minimal polynomial, and the product of all invariant factors becomes the characteristic polynomial.

Dummit & Foote, §12.1 — Theorem 1 and Corollary 2: a PID is Noetherian, and every submodule of a finitely generated module over a Noetherian ring is finitely generated; Proposition 3 bounds the rank of a submodule of a free module. ↩
Dummit & Foote, §12.1 — Theorem 4: the stacked (or aligned) basis theorem for a submodule of a finite-rank free module over a PID, with the divisibility chain $a_{1} ∣ a_{2} ∣ \dots ∣ a_{m}$ . ↩
Dummit & Foote, §12.1, Exercises 16–19 — the relations-matrix computation of the invariant factor decomposition over a Euclidean domain: row operations change the generators of $ker π$ , column operations change the basis of $R^{n}$ , and the reduction terminates in a diagonal matrix whose entries satisfy the divisibility chain. The diagonalization itself is the Smith normal form of §12.2, Theorem 21. ↩
Dummit & Foote, §12.1 — Theorems 5, 6, and 7: invariant-factor and elementary-divisor existence, and the primary decomposition of a torsion module into $p$ -primary components. ↩
Dummit & Foote, §12.1 — Theorem 9 and Corollary 10: uniqueness of both decompositions via reduction modulo a prime, and the deterministic conversion between invariant factors and elementary divisors; Corollary 11 specializes to $R = Z$ . ↩