The Structure Theorem for Finitely Generated
Modules over a Principal Ideal Domain
Dwight B. Thieme
University of Missouri - Columbia
Conference on Fabulous Presentations, 2010
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Outline
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is the Structure Theorem?
If M is a finitely generated module over a principal ideal domain
R, then:
M∼ = R f ⊕ R/(d1 ) ⊕ R/(d2 ) ⊕ · · · ⊕ R/(dn−f )
di 6= 0 or 1 and di |di+1 for 1 ≤ i ≤ n − f
The ideals (di ) are unique and the elements di are unique
up to multiplication by some unit.
This just says that M is isomorphic to a unique direct
product of cylcic modules R/(di ) an a free module R f .
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is the Structure Theorem?
This is a little mysterious. But since every abelian group (and
hence, every finitely generated abelian group) G is a module
over Z, we can restate the Structure theorem for this special
case in a more familiar way:
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is the Structure Theorem?
If G is a finitely generated abelian group, then:
G∼ = Zn ⊕ Zk ⊕ Zk ⊕ · · · ⊕ Zk
1 2 t
The rank of G is n ≥ 0
ki 6= 0 or 1, and ki |ki+1
This just says that a finitely-generated abelian group is the
direct sum of a free abelian group of finite rank and a finite
abelian group, each unique up to isomorphism.
The finite abelian group is just the torsion subgroup of G.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is Torsion?
Let R (or Z) be a principle ideal domain and let M (or G) be
a finitely generated R-module. If {g1 , g2 , . . . , gn } is a set of
generators for M, then there exists a surjective R-module
homomorphism:
ϕ : Rn → M
n
X
(r1 , r2 , . . . , rn ) 7→ ri gi
i=1
Let K be the kernel of ϕ. Since ϕ is surjective, M is the
image of ϕ and by the first isomorphism theorem:
M∼
= R n /K
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is Torsion?
Let R (or Z) be a principle ideal domain and let M (or G) be
a finitely generated R-module. If {g1 , g2 , . . . , gn } is a set of
generators for M, then there exists a surjective R-module
homomorphism:
ϕ : Rn → M
n
X
(r1 , r2 , . . . , rn ) 7→ ri gi
i=1
Let K be the kernel of ϕ. Since ϕ is surjective, M is the
image of ϕ and by the first isomorphism theorem:
M∼
= R n /K
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is Torsion?
n
P
If (r1 , r2 , . . . , rn ) ∈ K , then ri gi = 0
i=1
An element m of an R-module M is called a torsion
element if rm = 0 for some nonzero r ∈ R
if R = Z so that an abelian group G is a Z-module, then the
torsion subgroup of G is:
{g ∈ G |g| < ∞, ie, ng = 0 for some n ∈ N}
More generally, the kernel K gives a set of relations among
the generators {g1 , g2 , . . . , gn }, and K is the relation
submodule of R n relative to the given generating set.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is Torsion?
Let {k1 , k2 , . . . , km } ⊆ R n be a generating set of K .
If ki = (ai1 , ai2 , . . . , ain ), we can form the matrix:
a11 . . . a1n
A = ... .. ..
. .
am1 . . . amn
A is the relation matrix over R relative to the generating
sets {g1 , g2 , . . . , gn } of M and {k1 , k2 , . . . , km } of K .
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Example
Let M = Z3 ⊕ Z4 . Then M is generated by m1 = (1, 0) and
m2 = (0, 1).
The relations are then 3m1 = 0 and 4m2 = 0.
Consider the homomorphism ϕ : Z2 → M,
ϕ(r , s) = rm1 + sm2
Ker (ϕ) = {(r , s) ∈ Z2 (r + 3Z, s + 4Z = 0)} =
{(3a, 4b) a, b ∈ Z}.
Thus, (3,0),(0,4) is an ordered generating set for Ker (ϕ),
with the relation matrix:
3 0
0 4
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Example
Let the abelian group G have the generating set {g1 , g2 }.
Suppose the relation submodule K is generated by
(8,0),(0,12) so that the relation matrix is
8 0
0 12
Then K relative to {g1 , g2 } is
K = {(a(8, 0) + b(0, 12)) a, b ∈ Z} = {(8a, 12b) a, b ∈ Z}
K is the kernel of the surjective map σ : Z2 → Z8 ⊕ Z12
So then Z2 /K ∼= Z8 ⊕ Z12 , G ∼ = Z2 /K , and G ∼
= Z8 ⊕ Z12
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Big Fact
From this example, it is easy to see that if the relation
matrix is diagonal, we can determine M explicitly as the
direct sum of cyclic modules.
Suppose that A is a relation matrix for an R − module M. If
there exist invertible matrices S and T such that:
a1 0 . . .
0 a2 0 . . .
SAT = ... ..
.
an
0 ...
then M ∼
= R/(a1 ) ⊕ R/(a2 ) · · · ⊕ R/(an ).
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
The Smith Normal Form
Suppose R is a principle ideal domain and let A be an m × n
matrix. A is in Smith normal form if there are nonzero
a1 , a2 , . . . , at ∈ R such that ai ai+1 for 1 ≤ i ≤ t − 1 and A is a
diagonal matrix with:
a1
..
.
at
A=
0
..
.
0
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
The Smith Normal Form
Since R is a PID, we can always put a matrix into Smith normal
form with the following operations:
Add an integer multiple of one row to another or one
column to another
Interchange any two rows or any two columns
Multiply a row by −1
These operations are performed by left or right multiplying by
an invertible elementary matrix E.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
The Smith Normal Form
We use these operations in the following algorithm:
Find an entry of A with the smallest absolute value, then by
permuting rows and columns move this entry to the upper
left corner of A. Now try to make all other entries in the first
row and the first column 0 by the three given operations. If
these operations can only give a nonzero entry in a row or
column,
this
entry must have an absolute value smaller
than a11 , since aij = q a11 + r and 0 ≤ r < a11 . Start the
process over by permuting this value to the
upper left
corner of A. Since these successive a11 ’s form a
decreasing sequence of positive integers, we only have to
start over a finite number of times.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
The Smith Normal Form
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
The Smith Normal Form
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is Torsion?
You can create overlays. . .
using the pause command:
First item.
Second item.
using overlay specifications:
First item.
Second item.
using the general uncover command:
First item.
Second item.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is Torsion?
You can create overlays. . .
using the pause command:
First item.
Second item.
using overlay specifications:
First item.
Second item.
using the general uncover command:
First item.
Second item.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is Torsion?
You can create overlays. . .
using the pause command:
First item.
Second item.
using overlay specifications:
First item.
Second item.
using the general uncover command:
First item.
Second item.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is Torsion?
You can create overlays. . .
using the pause command:
First item.
Second item.
using overlay specifications:
First item.
Second item.
using the general uncover command:
First item.
Second item.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is Torsion?
You can create overlays. . .
using the pause command:
First item.
Second item.
using overlay specifications:
First item.
Second item.
using the general uncover command:
First item.
Second item.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
What is Torsion?
You can create overlays. . .
using the pause command:
First item.
Second item.
using overlay specifications:
First item.
Second item.
using the general uncover command:
First item.
Second item.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Make Titles Informative.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Make Titles Informative.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Make Titles Informative.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Make Titles Informative.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Make Titles Informative.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Make Titles Informative.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Make Titles Informative.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Make Titles Informative.
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
Every finitely generated module M over a principal ideal
domain R is isomorphic to a unique decomposition of the
form:
R f ⊕i R/(di ) ≡ R f ⊕ R/(d1 ) ⊕ R/(d2 ) ⊕ · · · ⊕ R/(dn−f ),
where di 6= 0 or 1, and di |di+1 . The ideals, (di ), are unique,
and the elements, di , are unique up to multiplication by
some unit.
Every finitely-generated abelian group, G, is a module over
Z and hence the Structure Theorem can be given in this
form:
G∼= Zn ⊕ Zk1 ⊕ Zk2 ⊕ · · · ⊕ Zkt ,
with rank n ≥ 0, ki 6= 0 or 1, and ki |ki+1 . This just says that
a finitely-generated abelian group is the direct sum of a
free abelian group of finite rank and a finite abelian group,
each unique up to isomorphism. The finite abelian group is
just the torsion subgroup of G. The rank of G is defined as
the rank of the torsion-free part of G; this is just the
number n in the above formulas. The first main message of
Dwight B. Thieme The Structure Theorem for Finitely Generated Modules over a Prin
