Similarity of matrices over a ring

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7 90 33 E 2 MCMAHON* E D I T H MARY S I M I L A R I T Y OF M A T R I C E S NORTHWESTERN

OVER

UNIVERSITY,

University Microfilms International

300 n . z e e b r o a o , a n n a r b o r , mi 4 8 io 6

A RING.

PH.D.,

1973

NORTHWESTERN UNIVERSITY

S I M I L A R I T Y OF M A T R I C E S O V E R A RING

A DISSERTATION S U B M I T T E D TO THE

GRADUATE

SCHOOL

IN P A R T I A L F U L F I L L M E N T OF THE R E Q U I R E M E N T S for the d e g r e e D O C T O R OF P H I L O S O P H Y F i e l d of M a t h e m a t i c s

By Edith Mary McMahon

Eva ns to n, June,

Ill ino is 1978

T a b l e of C o nt en ts Page I n t r o d u c t i o n ................. Chapter

1:

1 to a M a t r i x ...........

The M o d u l e A s s o c i a t e d

1.

D e f i n i t i o n of

the M o d u l e .........................

2.

The A n n i h i l a t o r

3.

Similarity

of

M

of M a t r i c e s

in

R [ A ] ................

and

Isomorphism

of M o d u l e s .......................................... 4. Chapter

Properties 2:

of the R in gs

Lattices

Ov e r O r d e r s

1.

Definitions

2.

The Jordan-Zassenhaus

3. Chapter

and

A

.........

...............

Re sul ts

11 12

17 19 25

..................

25

C o n d i t i o n ..... ...........

27

G e n e r a o f L a t t i c e s .................................

33

3:

40

The

and B a si c

A

11

Latimer-MacDuffee Correspondence

. . .

1.

T h e C l a s s i c a l M e t h o d ..............................

40

2.

The Relationship

47

Chapter 1.

4:

of M o d u l e s

Block Triangular

Transforming

Form.

a Matrix

and

Ideals. . . .

.......................

into B l o c k T r i a n g u l a r

F o r m ................................................. 2.

Similarity

of M a t r i c e s

5:

Determination

of S i m i l a r i t y .................. i

53

in B l o c k T r i a n g u l a r

F o r m .................................................. Chapter

53

63 69

Page 1.

Introductory

2.

Form

1:

R e m a r k s ...........................

£( X)has d i s t i n c t

.

69

irreducible

f a c t o r s .................................................. 69 3.

F o r m 2: same,

All f(A)

the

roots

of

the

(A - a ) n

.............................. 71 - a) ,and

Fo rm

3:

f(A)

= g(A)(A

5.

Form

4:

f(A)

= g(A )h( A) and

6.

are

=

4.

degree 2

f(A)

g(a)

h(A)

^ 0

.

81

has

............................................... 85

S u m m a r y .................................................. 93

B i b l i o g r a p h y ....................................................... 98 V i t a ................................................................ 101

ii

INTRODUCTION

T he p u r p o s e aspects We

of this

of s i m i l a r i t y

first

dissertation

of s q u a r e m a t r i c e s

di s c u s s

the n u m b e r

w h e n this n u m b e r

is finite.

We then

whether

A'

R

whether that

similar th er e

AP

e x is ts

= PA'

approaches

to

over

.

.

a ring

turn to the

Th a t

is,

R

specific A

and

we w i s h to d e t e r m i n e P

ov er

R

fi rst d e s c r i b e our m e t h o d s

the se p r o b l e m s ,

.

shox^ing

two squ ar e m a t r i c e s ,

an i n v e r t i b l e m a t r i x

We wi l l

over

two

of s i m i l a r i t y cl as se s,

p r o b l e m of d e t e r m i n i n g , are

is to s tu dy

suc h and

and then m a k e a c o m p a r i s o n of

our w o r k w i t h r e s u l t s p r e v i o u s l y o b t a i n e d by others. The

first

two c h a p t e r s

determining when

the n u m b e r of s i m i l a r i t y cl as s e s

We b e g i n by c o n s i d e r i n g with

identity.

ea c h e l e m e n t elements Ma

in

from R [A ]

therefore of

A

, we

We A

of R

is

.

Rn

an R [A ] -module,

,the ri ng

After proving

the nu l l ideal

M

We n o t e

finitely generated

of that

of the

n x n

Ma

ring,

that,

by its

important

R

,

, with

the a n n i h i l a t o r of matrix

A

and f(A)

as a A = R [A ] / (f (X) ) - m o d u l e

A-module which be

is finite.

matrices with

the c h a r a c t e r i s t i c p o l y n o m i a l ,

consider

a fact w h i c h wi l l

an a r b i t r a r y c o m m u t a t i v e

associate

contains

ob v i o u s way.

are p r i m a r i l y c o n c e r n e d w i t h

construction,

MA

,

in the is a

is free as an R - m o du l e ,

in the sec on d cha pter.

2

A necessary of

Rn

to be

condition

similar

teristic polynomial, MA

a nd

prove ,

Mg

is that

they have

f(A)

Thus,

re s p e c t i v e l y ,

in T h e o r e m 1.5,

if and onl y T he re su l t s

if th ei r

.

the

also

consider

to tal

subring the

quotient

of

A

in t e g r a l

integral

c h a p t e r pr e s e n t s

up

of

a p p r o a c h we

orders.

An

containing

Q

.

We

and

R-order

in

in

A A

A

order

a n d that

A

of

in Rn

A .

of m a tri x.

, where A

ov e r

R

in cl u d e

Q

is

is a .

is sho wn to be

Further, the

to be a s e p ar ab le Q - a l g e b r a .

a s s o c i a t e d to m a t r i ce s.

is a D e d e k i n d d o m a i n w i t h

A

is a f i n i t e - d i m e n s i o n a l

is to use S

in

A

is a su br in g

su ch that

an R-module. MA

Q-algebra.

the t h e o r y of la t t i c e s

is an S - m o d u l e w h i c h as

ideal

and

. We give n e c e s s a r y and

g e n e r a t e d R - m o d u l e w h i c h co n t a i n s

torsion-free

We

R

the id en ti ty ,

S -lattice

similar

definitions

show that

R

of the A - m o d u l e s

take

are

is c o n c e r n e d w i t h the c l a s s i f i c a t i o n ,

the c h a p t e r

fie ld

R

is in te g r a l

for

second chapter

to i s o m o r p h i s m ,

quotient

A

the null

A

of

conditions

Throughout

The

closure

W e then

to in lat er chapters.

ring of

,

are A - i s o m o r p h i c .

A = Q[A ]/ (f (A ))

a n d that

closure

sufficient The

the ring

B

the same c h a r a c ­

two such m a t r i c e s

a d i s c u s s i o n of h o w to d e t e r m i n e We

and

the a s s o c i a t e d m o d u l e s ,

associated modules

referred

A

are bo t h A-modu le s.

that

rest of the first w h i c h are

for two ma tr ice s,

We

S

of

A

over

,

is a f i n i t e l y

a Q-basis

of

A

.

is f i n i t e l y g e n e r a t e d sh ow that

is a A - l a t t i c e

A

An and

is an R-

for a ny

el e m e n t

Historically, content

of a l g e b r a i c n u m b e r

r i n g of r a t i o n a l O'

, of the

number

0

of the

Dedekind

it to be

A 1

, was

i n te ge rs

st ud i e d

in the

-order,

d e f i n e d to be a s u br in g 0

studied

of an a l g e b r a i c the

and a m u l t i p l e

t h e o r y of l a t t i c e s

in r e p r e s e n t a t i o n as we l l

as

group

ri n g

RH

theory.

Curtis

to r e p r e s e n t a t i o n

ideal of the

cl as s of

o v e r or de r s Roggenkamp

and Reiner

theory.

has

H

is an R - o r d e r

in

also be e n used

and H u b e r - D y s o n

[6 ] stress

If

This

subject

ment

for the s p e c i a l

survey

[24].

is d e v e l o p e d

of i n t e g r a l

In this case,

matrices

and

an

in

this

is a finite QH

[6 ],

case w h e n

.

we

integral

1

representation

then

group,

[22].

Thus,

of s i m i l a r i t y c l a s s e s

F r o m our w o r k

in C h a p t e r

of

is g i v e n

H

in

t h e o r y can be relationship

and

If

the

.

f(A)

~J_ [0 ] . defines

f(A)

[29]. fo un d

in

between is an

is a root of

0

su c h that of

P _1A P

irreducible representations number

A

representation

invertible matrix,

R = 1

[X]

approach

E a c h R H - l a t t i c e wil

representations. in

[25,

s u m m a r y of the d e v e l o p

see a d i r e c t

= 0 , th e n e a c h m a t r i x

irreducible

A

representation

irreducible polynomial f(A)

O'

class n u m b e r

a f f o r d a set of R - e q u i v a l e n t R - r e p r e s e n t a t i o n s

A

for the

. The

26]

/

[7]

finite

first

theory.

integers,

rings

field.

showing of

R-orders were

If

= 0 P

define

is an

an e q u i v a l e n t

the n u m b e r of i n e q u i v a l e n t of

~][ [0 ]

is the

same as the

of m a t r i x ro ot s of

1, we k n o w this

f(A)

= 0

to be the n u m b e r

.

of i s o m o r p h i s m c la ss es ciated

s h o w that this n u m b e r

of A - l a t t i c e s quotient

integral

finite when Q

is a g l ob a l

as s t a t e d

asso­

in

[23],

f i el d and that

holds

R

in

A

f(A)

.

a separable

for A - l a t t i c e s , in

we

A

using

is the

s h o w that the

of m a t r i c e s w i t h c h a r a c t e r i s t i c

approach

to

the q u e s t i o n of the n u m b e r of

of A - l a t t i c e s

finite

Th e n u m b e r of c l a s s e s

and can be g i v e n

class n u m b e r of a m a x i m a l

R-orde r.

as

th e y a p p l y to the A - l a t t i c e s

We

t h e n use

these

is g i v e n by J a c o b i n s k i

the c l a s s i f i c a t i o n of A - l a t t i c e s

a n d r e s t r i c t e d genera.

results

matrices

are s i m i l a r

lattices

are

in terms

of

We p r e s e n t

the

into

is ideal

these results

a s s o c i a t e d to m a t r i c e s .

in an e x a m p l e

if a n d o n l y

sh o w i n g w h e n

two

if t h e i r a s s o c i a t e d A-

in the sa m e genus.

W e n o w turn given matrices the thesis,

is

is finite.

He d e v e l o p s

s h o w n to be

A

the J o r d a n - Z a s s e n h a u s

Thus,

of

isomorphism classes

to the p r o b l e m of d e t e r m i n i n g w h e t h e r

ar e similar.

we a s s u m e that

an a l g e b r a i c n u m b e r insuring

is a D e d e k i n d d o m a i n w h o s e

closure

of s i m i l a r i t y c l a s s e s

[10,11].

of i s o m o r p h i s m c l a s s e s

R-order

A different

cular

la t t i c e s

that the u n i q u e m a x i m a l

polynomial

genera

R

We do this by s h o w i n g

condition, fact

is

field

Q-a lg e b r a .

number

~jj [A]/(f(A))

to such m a t r i c e s .

W e will

the

of

unique

Throughout R

f ie ld w i t h

factorization.

situation depends

on the

this p o r t i o n

of

is the ring of i n t e g e r s c la ss n u m b e r one, The m e t h o d u s e d

two

of

thus in a p a r t i ­

fo r m of the c h a r a c t e r i s t i c

5

polynomial wil l

f(A)

tell us A

fa ct o rs

of f(A)

is

have

o f classes.

ideal w i t h a nd onl y T he

s p ec ia l

fa ct or s w i t h ea c h matrix .

3.

ba sis T he

for

polynomial, ~]L [C]

,

, where

Taussky

[30]

C

of

specializes

f(A)

ciated

to the m a t r i x

the ring of in t e g e r s number

one.

ideals

resulting

in f i n i t e this

distinct

are

then are

correspondence

In

an

similar

if

in the same class. is p r e s e n t e d

are g e n e r a l i z a t i o n s [15]

f(A) ideals

as

they

establish

in the ring of p o l y n o m i a l s

In this

the m o r e

out

of

characteristic

case,

ge ne r a l

f(A) to the the

. case

ideal

asso­

We p r e s e n t

case w h e n

of an a l g e b r a i c n u m b e r

R

is

fi el d of class

the c o n n e c t i o n b e t w e e n

from the L a t i m e r - M a c D u f f e e defined

Q-algebra

, we can a s s o c i a t e

the c o r r e s p o n d e n c e

We also p o i n t

a nd the m o d u l e s

or has

is s p e c i f i c a l l y de scr ibe d. for

irreducible

between similarity classes

is irr edu ci bl e.

re su lt s

and

illustrates

is the c o m p a n i o n m a t r i x of

\\rhere

all of th ese

R

g i v e n th ere

w h i c h hav e

and cl a s s e s

R[A]

5.

of L a t i m e r an d Ma cD u f f e e .

in

the

a separable

case w h i c h

ideals

re s u l t s

= 0

we m a y have an

over

this

f(A)

in

of s i m i l a r i t y classes.

The matrices

a one-to-one correspondence matrices

is

f(0) f 0

if the a s s o c i a t e d

in C h a p t e r

A

in C h a p t e r

is i r r e d u c i b l e

theoretical

of th ose

A

of

f(A)

n o t e that w h e n

are not d i s t i n c t ,

in d e ta il

f(A)

We

a fini te n u m b e r

number

is d i s c u s s e d

of the roots

are di st in ct ,

fa ct or s

irreducible

f a c t o r i z a t i o n of

separa b le.

If the

If

The

the n a t u r e

whether

and we wi l l

.

in C h a p t e r

1.

We

correspondence

show

that

an

6

ideal

associated

A-homomorphism matrix

of

to the m a t r i x

fr o m

f(A)

MA

to

A

*MC

will where

case w h e n integers This

to the ca s e w h e n

of an a l g e b r a i c n u m b e r

polynomial

the m a i n d i a g o n a l wi l l

matrices

R

[20]

is the

f r o m the ring of

fi el d w i t h class n u m b e r

m a y be t r a n s f o r m e d

along

zeroes

has r e p e a t e d

f(A)

in b l o c k t r i a n g u l a r

be

companion

that e v e r y m a t r i x w i t h c h a r a c t e r i s t i c

is

f (A )

the

shows

which

of

f(A)

first g e n e r a l i z e a r e s u l t of

R = ]J_

result

is

4 w e p r e s e n t an a p p r o a c h to the q u e s t i o n of

s i m i l a r i t y w h i c h m a y be u s e d w h e n We

C

to a

.

In C h a p t e r

factors.

correspond

as its

form.

into a s i m i l a r m a t r i x Th a t

is,

characteristic polynomial,

form,

reduced

the

given

the q u e s t i o n of to a c o n s i d e r a t i o n of

the s i m i l a r i t y of c o r r e s p o n d i n g blocks. exactly how a matrix

fa ct o r

a n d t h er e wi ll

By t r a n s f o r m i n g

into b l o c k t r i a n g u l a r

s i m i l a r i t y can be p a r t i a l l y

ea c h b l o c k

have an i r r e d u c i b l e

b e l o w these blocks.

one.

We also

ca n be t r a n s f o r m e d

specify

into b l o c k t r i ­

a n g u l a r form. Using

this

two m a t r i c e s , similar.

result, A

We do

and

we d i s c u s s

A 1 , in b l o c k

invertible matrix

A

A'

are

are

the b as is

P

similar via

of our w o r k

In that ch ap te r,

triangular

so by g i v i n g c o n d i t i o n s ,

that an and

h o w to d e t e r m i n e w h e t h e r

we

must

in T h e o r e m 4.6,

satisfy

A P = PA'

.

specify procedures

in o r d e r tha t

These

t h r o u g h o u t mos t

fo r m are

conditions

of C h a p t e r

to be u s e d

5.

in de ter-

mining whether

A

and

special methods

for s e ve ra l p o s s i b l e

teristic polynomial, The first irreducible results

of C h a p t e r

and

are

f(0) = 0

f(0)

, we n o t e

We p r e s e n t

forms of the c h a r a c ­

f(A)

are

has

In this

to a s s o c i a t e

If

that

is w h e n f 0 .

ideals

similar.

similar.

, individually.

3 are u s e d

If th es e

the m a t r i c e s but

f(A)

are

fo r m c o n s i d e r e d

f a ct or s

e a c h matrix.

A'

distinct

case,

the

an ideal w i t h

in the same class,

f(A)

has d i s t i n c t

then

factors,

a c o m b i n a t i o n of m e t h o d s

is to

be used. Secon dl y, has an n - f o l d case, f(A)

f(A) =

we consider ro o t

= An

(A - a ) n

in

.

R

.

the m i n i m u m

and

A'

, then we

= PA*

gi v e

.

When

We in this

case.

ity c l a s s e s

first

, i.e.

look at the n i l p o t e n t

We p r o v e

that

case if

for

f(A)

have n e c e s s a r y and s u f f i c i e n t

f(A)

P

is not shows

that

A

T h e r e wi l l

of m a t r i c e s

f(A)

l a t e r that the g e n e r a l

must

satisfy

case,

the d e t e r ­ = An

is

for

A

conditions

in o r d e r

that

the m i n i m u m p o l y n o m i a l ,

that these

but n o t n e c e s s a r y

also n o t e

(A - a ) n

and the c h a r a c t e r i s t i c p o l y n o m i a l

an e x a m p l e w h i c h

sufficient,

=

ca n be r e d u c e d to this

w h i c h an i n v e r t i b l e m a t r i x AP

We

We p r o v e

m i n a t i o n of sim il ar it y. both

f(A)

conditions

we

are

for similarity.

is not

a se p a r a b l e

Q-algebra

be an i n f i n i t e n u m b e r of s i m i l a r ­

having

characteristic polynomial

over

n u m b e r of n o n - a s s o c i a t e s .

As

f(A)

= An

as m i n i m u m and

any ring w i t h an an example,

infinite

we l o o k at the

8

case w h e r e

f(A)

= A2

and

specify a representative W e nex t g(a) 4.6

f 0 .

lo o k at the The

in this

must have

is m o r e

f(A)

distinct

may

has

consider

with

g(0)

similarity given

last

g ( A) h( A) modify

with

We

for

h(A)

case

followed

in d e t e r m i n i n g

We

examine

described whether indicate

fo r m

the

the c o n d i t i o n s

4.6

g(A)

that

if

= 0 , we • g(A) for the

similarity. is

f(A)

in this

=

We a g a i n case

to o b ta in the

specia l

further modify

and

h(A)

these

.

a s u m m a r y of the p r o c e d u r e whether

two m a t r i c e s

for m of forms

the g e n e r a l p r o c e d u r e .

to be

similar.

a nd the

of the m a t r i c e s .

of C h a p t e r

similar.

are

f(A)

using a combination

se c t i o n s are

P

a d e t e r m i n a t i o n m a y be p o s s i b l e

factored

in stages,

the m a t r i c e s

= A

We also c o n s i d e r

block triangular

in p r e v i o u s

f(0)

f(A)

is also q u a d r a t i c and case,

in T h e o r e m

in c o m b i n a t i o n w i t h

of T h e o r e m

the p o l y n o m i a l s

then proceed

and

fo r m c o n s i d e r e d

by e x a m i n i n g

corresponding

the

with

We also n o t e

to d e t e r m i n e

In this

first

- a)

a quadratic polynomial.

we p r e s e n t

we

the fo r m w h i c h

fa cto rs

section

conditions.

Fin a l l y ,

= g(A)(A

so that

for si m il ar it y.

g(A)

exa mpl e,

similarity given

then m a y use

in this

the c o n d i t i o n s

ca se w h e n

f(A)

as h a v i n g

individual

new conditions

form

irreducible

Latimer-MacDuffee method T he

In this

easily determined.

f(A)

f 0 .

.

of each s i m i l a r i t y class.

conditions

are m o d i f i e d

R = 1

We

of the m e t h o d s

5,

An example We no t e here

to d e t e r m i n e is g i v e n to that

all

cases

9

for

n £

la rg er

5 n

can be h a n d l e d u s i n g

, a r e d u c t i o n to a si mp le r case

We will n o w d e s c r i b e an d p r e v i o u s

results.

chapter

is p r i m a r i l y

for

Theorem

1.5,

w h i c h shows

c o r r e s p o n d to s i m i l a r m a t r i c e s re sults.

Our w o r k

ing re su l t s s tu dy

the

be

associated

the n u m b e r

T he

fi rst

y i el ds

of s i m i l a r i t y cl as s e s

s e c t i o n of C h a p t e r result

on a s s o c i a t i n g R

Latimer-MacDuffee

the

s e c t i o n of that

chapter.

4 is al s o

that m a t r i c e s

number

that

there

statement can thus

this

is

one.

ideals

of an

We d e ve l o p

associated

T he p r e s e n t a t i o n

in the

to se co n d

in the

fir st pa rt

a g e n e r a l i z a t i o n of a k n o w n result,

ca n be t r a n s f o r m e d result

to an in teg ral

is the ring of i n t e g e r s

matrices

R

this

is fi ni t e

an ideal

b e t w e e n the A - m o d u l e s

ca se w h e n

The

to the

3 is the g e n e r a l i z a t i o n

the r e l a t i o n s h i p

extend

of r e l a t ­

We do

the r e su lt

f i e l d w i t h class n u m b e r

We

A-modules

over o r de rs

algebraic number

of C h a p t e r

informa­

e x t e n s i o n of k n o w n

of .isomorphism classes.

to the c a s e w h e r e

and

first

stated.

of a c l a s s i c a l matrix

is a slight

to ma tr ice s.

t h e o r y of l a tt ic es

explicitly

isomorphic

in the se co n d c h a p t e r c o n s i s t s

a f i ni te n u m b e r

of w h e n

that

in the

of b a c k g r o u n d

the t h e o r y of lat ti ce s

of A - m o d u l e s

because are

fr o m

b e t w e e n our

The m a t e r i a l

the p u r p o s e

For

is o f t e n poss ible.

the r e l a t i o n s h i p

re su l t s

tion.

this p ro ce du re .

into b l o c k t r i a n g u l a r

from the case w h e n

the ring of i n te ge rs

fi el d w i t h cla ss

n u m b e r one.

R = ~JJ_

form.

to the

of an a l g e b r a i c The r e m a i n d e r

of this

10

c h a p t e r g ive s

our

i n t e r p r e t a t i o n of this result,

h o w it can be u s e d In C h a p t e r

5 we

in the

ea c h o f these Thus, extended ring

d e t e r m i n a t i o n of sim il ar it y.

consider various possible

g i v i n g m e t h o d s to be used

sh owi ng

for d e t e r m i n i n g

forms

for

f(A)

similarity

in

cases.

in the

results

of i n t e g e r s We

latter part for

R = ~]j_

of the thesis, to the case w h e n

o f an a l g e b r a i c n u m b e r have

we have

number

one.

then us e d t he se

method

of d e t e r m i n i n g w h e n two m a t r i c e s

R

field w i t h

resul ts are

is the class

to de ve l o p similar.

a

,

Chapter

1

The Module Associated

1.

D e f i n i t i o n of the M o d u l e Throughout

rin g w i t h

this

ide ntity.

chapter, For

A

R

the ri n g of p o l y n o m i a l s

fro m

.

R

Rn

wi ll

h a v i n g en tr i e s h(A)

= r m Am +

h(A) in

from ...

denote R

.

from

R Let

+ r xA + r 0

denote

a commutative

X

A

is

is an

R [A ]

with coefficients

of

n xn

matrices

an el e m e nt

of

el em en t of

r mA m + in

...

A

Rn

R[A]

+ r xA

with

and , then

+ r 0I

coefficients

. A

=

(a^j) is

be an el e m e nt of f(A)

.

{ e j , e 2 , . .. ,e n } .

Let

M

We d e f i n e

Rn

whose

character­

be a free R - m o d u l e w i t h the a c t i o n of

A

on

by

A • v = A

• v

to all

e le me nt s

by

A *

We d e f i n e for all h(A) is

If

, the rin g of p o l y n o m i a l s

istic p o l y n o m i a l R-basis

in

the ring

will d e n o t e the p o l y n o m i a l R [A ]

will

an i n d e t e r m i na te ,

de n o t e s

M

to a M a t r i x

of

v e M R[A]

therefore

e• = i

n 7 a ••e • . .L _ 11 j j= l

the a c t i o n of

A

.

extend s

This

via

action

h(A)

on

• v = h(A)

an R [A ] -module. 11

M

• v

.

We

see

that

M

12

It is c l e a r that si nc e Rn

is not

free as an R [ A ] - m o d u l e

the C a y l e y - H a m i l t o n T h e o r e m h o l d s

[12,

p 101].

R[A]-basis phism

Let

F

T :F -* M

and d e n o t e

for m a t r i c e s

in

be a free R [X ] -modu le w i t h

{ u 1 ,...,un } .

D e f i n e an R [A ] -m od ul e h o m o m o r ­

by

T(

map,

M

its

I h i (A) ui) = I i=l i=l

ke r n e l

by

Kcr T

.

h i (A)ei

T

is c l e a r l y an onto

so w e have

M s F/Ker T

Note

that

M

is

.

a finitely generated R[A]-module which

is

free as an R-m odu le .

2.

The Annihilator Let

A

polynomial section we annihilate We in

A

n x n If

be

M

an el em e n t

f(A)

R[A]

of

Rn

f irs t n o t e

{h(A)

that

with coefficients

+

...

R n [A]

, it c o r r e s p o n d s

entry

is the p o l y n o m i a l is the

e R[A] |h (A)

in

.

In this

R[A]

• M = 0}

which

.

> the ring o f p o l y n o m i a l s

from

Rn from

+ AiA

and

R[^]n

R[A]

ar e

+ A0

Ak

isomorphic.

R[A]

a m Am + a m _ i A m 1 + e n t r y of

» the ring of

is an e l e m e n t

to the m a t r i x of

(i,j)

M

R n [^]

m a t r i c e s w i t h e n tr ie s

ak

with characteristic

the set of p o l y n o m i a l s

, i.e.

A m Am + A m _ i A m _ 1

where

in

and a s s o c i a t e d R [ A ] - m o d u l e

specify M

of

,

...

whose

of (i,j)

+ axA + a 0

k = 0,1,...,m

.

13

We wi l l

identify

nomial ,

h(A)

m a t r i x of

, of

R [AJ n

T h e o r e m 1.1 h(A)

these

R [A ]

Let

Suppose

R [ A ] - a c t i o n on

h(A)

h(A)

M

if

h(A)

= 0

M

on

• M = 0 . M

, we

see

T h e set R[A]

, called

U si ng that

f(A)

= h(A) •

if

R[A]

h(A) =

.

Th e n

0 . the

• M

i m pli es

.

= 0 . This

ideal

R[A]

means

that

[]

= 0} is

of the m a t r i x

[16,

the C a y l e y - H a m i l t o n

Thus

n

vi a

y(h(A ))

= R[A]/„

the anni-

polynomial.

of the on t o ma p

isomorphism

R [A]

is

.

= h(A)

.

of

p 162].

is a n o n - z e r o

(hence no n- ze ro )

-> R[A]

A

ideal

.

.

of

A

an

ideal

Since

A

= 0.

the n u l l

e n

the

h(A)

e R [A ] |h(A)

is the k e r n e l

we h a v e

M = 0

.

h(A)

this

f(A)

y:R[A]

Therefore,

of

the d e f i n i t i o n of the a c t i o n o f

is a m o n i c

-n

as a s c al ar

By the d e f i n i t i o n of

M

= {h(A)

in

T h e o r e m holds,

that

• M

the null

M

a poly­

, we have

s h o w n that

h i l a t o r of

note

.

suppose

rr

T h u s w e ha v e

s in ce

and o n l y

annihilates

Conversely, h(A)

Thus,

m a y be c o n s i d e r e d

be an el e m e n t

h(A)

Thus,

rings.

.

annihilates

Proof

isomorphic

ideal We

14

Let

(f(A))

de n ot e

the p r i n c i p a l

ideal

of

R[A]

g e n e r a t e d by the c h a r a c t e r i s t i c p o l y n o m i a l

of

A

(adj(AI-A))

AI

- A

denote

the a d j o i n t m a t r i x of

.

Let .

We m a y

wr i t e

f(A)

We use

this

following

Theorem g(A)

c on c e p t

1.2

[16]

r ight

such

Suppose (AI

f(A)

zero d i v i s o r f(A)

Let

g(A)

if there

(*)

exi st s

- A)

is satisf ie d.

.

in the

R [A ] .

an m a t r i x

Th e n

K(A)

in

.

M u l t i p l i c a t i o n on the

yields:

= K(A)f(A)(AI

- A)

= K( A)( AI

- A) f (A )

.

is a m o n i c p o l y n o m i a l in

R[A]

in

d i v i d e b o t h si des

R[A]n

.

Thus,

we may

, it is not

a by

, so

g(A)

Therefore

g(A)

= 0

Conversely, th er e

be an el em en t of

g (A) (adj (AI-A) ) = K(A) f (A)

by

- A)

that

g( A)f(A)

Si n ce

(adj(AI-A))(AI

of c h a r a c t e r i s t i c p o l y n o m i a l

if and only

(*)

Proof

=

theorem.

e 77

R[A]n

= de t(AI-A)

is a m a t r i x

= K ( A) (A I

- A)

.

and

g(A)

e

suppose

g(A)

= 0 .It t h e n

K(A)

g (A)

in

n

R[A]n

= K ( A)( AI

su ch that

- A)

.

fo l l o w s

t hat

15

Multiplication

on the right

by

adj(AI-A)

then yiel ds

(*),

as desired.

[]

T h e o r e m 1.2 elements

of

su gge sts .

ti

adj (AI-A)

, that

e n t r y of

(AI

Let

is,

- A)

a m e t h o d of d e t e r m i n i n g j (A)

A^j(A)

.

denote

is

the

the

(i,j) en try

the c o f a c t o r

of the

of

(i,j)

Th e r e s u l t of T h e o r e m 1.2 can be

restated: The p o l y n o m i a l on l y n2

if

g(A)

g(A)

e R [A ]

satisfies

i

1 , 2 , . . . ,n , j every polynomial

uniquely

as

g (A)

the d e g r e e

of

a monic polynomial elements

of

less

Theorem in terms

than

cn-

p(A)

[32,

n

1.2

ideal

of

mod

g(A)

s y s t e m of

S

. m a y be w r i t t e n

+ q (A) f (A)

is less 30].

(f (A)) ,

e R[A]

than

Thus,

n

, s in ce f(A)

several

s ol uti on s,

of

s y s t e m of co ng ru en ce s. straightforward consequences

of an ideal.

be a c o m m u t a t i v e ri n g w i t h S

is

to d e t e r m i n e the

find all n o n - z e r o

, to the ab ov e

of the r a d i c a l

an

p

also has

Let

= 0

= p (A)

, we w o u l d

D e f i n i t i o n 1.1 and

if and

following

1 , 2 , . . *,n

Fur th er ,

degree

the

~n

congruences

g ( A ) A i j (A)

where

is in

.

The r a d i c a l

of

identity

, r( 0}

of

S .

n

, of

.

A

is

. t~i

.

As

in the

theorem,

su c h that

- A)

.

that

det( g( A) )

= d e t( K( A))

• de t(AI-A)

,

or

g (A)

so

g (A)n e

Corollary Proof som e

(f (A))

1.4

As

(f(A))

Co nv e r s e l y , 77

, (f(A))

Thu s

h(A)

5

g(A)

77

e r(f(A)) .

, that

in the p r e v i o u s so

g(A)

suppose so

e r (77 ) .

• f(A)

£ ( 7 7 ) = r(f(A))

g(A) e r(77)

m > 0 .

g ( A ) mn s

and

[16]

Suppose

= det( K( A) )

h(A)

.

is, g ( A ) m

e

cor ol la ry ,

£ r(f (A)) h(A)

[]

for

we have

.

e r(f(A))

e(f(A))

77

5

77

. Since for some

f(A) m

e >0 []

.

17

It is well ^

k now n

[2, p 9]

is the i n t e r s e c t i o n of the p r i m e

The last c o r o l l a r y t h e r e f o r e structures are

of the

rings

tells

prime

r(f(A))

ideal,

=

containing

us that the p r i m e

R[ A] /( f( X) )

(f(A))

and

R[A]

^

.

ideal

= R [X ]/7^

which R[A]

impli es .

, for

5

that

In this

=

r(f(A))

(f(A))

Throughout

having

of

M

in

MA

and

Mg

and

respectively.

Mg

, we m a y c o n s i d e r b o t h

R

we c o n s i d e r

two m a t r i c e s ,

Si nc e MA

A

as c h a r a c t e r i s t i c p o l y n o m i a l .

the R [A ] -modules

in the ob v i o u s way.

are s i m i l a r over

I s o m o r p h i s m of M o d u l e s

f(A)

A

modules

c n ,

(f(A))

= R[A]/„

and

this section,

Rn

We d e n o t e by B

is a

case \\re have

S i m i l a r i t y of M a t r i c e s

, of

(f(A))

is the a n n i h i l a t o r

R[A]/(f(A))

B

example when

we have

n

an d

ideals

of an ideal

identical. If

3.

that the radic al

f(A) and

if and only

annihilates

Mg

We wi l l

a s s o c i a t e d to

as

MA

and

R[A]/(f(A))-

sh o w that

A

and

B

if their a s s o c i a t e d m o d u l e s

are R [A ] / ( f( A) ) - i s o m o r p h i c .

Definition say

that

exists R

and

1.2 A

For two

is s i m i l a r

a matrix AP

= PB

P .

of

el em e n ts

A

to

B

over

R

such that

and R

B

of

Rn

if and o n l y det(P)

, we if there

is a unit of

18

We m a y of

also expres s

R[A]- actions.

an d by

TB

v e Ma

and all

w e Mg

A

is s i m i l a r

exis ts

an R - m o d u l e

(y o T a )

If

P = (P j j )

A

a nd B

to

in terms

.

That

is,

MA

for all

,

and

B

TB • w = A • w

over

isomorphism

• v =

of s i m i l a r i t y

the R [ A ] - a c t i o n on

on Mg

• v = A • v

Thus,

co n c e p t

Den ot e by T A

the R [ A ] - a c t i o n

TA

a c t i o n of

the

if and on l y

if there

y :MA ->■ M B such that

(T b o y)

is the m a t r i x

R

.

• v

for all

w h i c h yi el d s

as

in the d e f i n i t i o n ,

P

on

th e n

v e MA

.

the s i m i l a r i t y

y

represents

of

the

MA n

1

for

the

R-basis

as a c h a n g e R-module

Theorem Ma

of basis

1.5 A Mg

is are

of

an R - i s o m o r p h i s m

first

for all

that

v

= A • (y(v))

is o m o r p h i s m ,

E

J=1

p ijei

MA

an d

to

isomorphic

e x is ts

• y(v)

MA

si mi l a r

Suppose

Tg

=

.

We m a y

consider

y

R - i s o m o r p h i s m on the u n d e r l y i n g

Proof

T B o y(v)

ei

{ e 1 } ...,en } of

structures

a nd

'

MB .

B

ove r

is

similar

y : M A -»■ M B

, it

if and on ly

if

R[A]/(f(A)) -modules. A

in

R

MA

.

B

su ch that

Since

is c l e a r

to

TA

that

, so t he re y o T A (v)

• v = A • v y

h e n c e an R [A ] / ( f ( A ) ) -isomor ph is m.

= and

is an R[A]-

19

Conversely, is om or p hi sm . denote R[A]

its

su pp os e

Y-’M a

as an

For any e l em e n t

image

in

-> R [ A ] /( f (A ))

h(A)

R[ A ]/ (f (A )) .

/ (f C^) ) “

e R [A ] , let

under

Th e a c t i o n of

h(A)

the c a n o n i c a l m ap

h(A)

on

MA

is,

by

de f i n it io n ,

h(A)

• v = h(A)

It is n o w c l e a r

that

v e MA y

module

structures

is an R - m o d u l e

are s i m i l a r o v e r

Properties Th is

involved

of

A

R

1.3

A

s u c h th a t

Definition

1.4

ring

O)

, we have

that

A

R-

an d

B []

A

and

in

If

R

The

of two

re s u l ts

la te r ch apters.

here

rings are

to

We w i l l not m a k e

or t h e i r a s s o c i a t e d m o d u l e s .

f o l l o w i n g de fi ni ti on s.

regular element ar

A

s e ve ra l p r o p e r t i e s

to m a t r i c e s

o n e r e g u l a r el ement, the u n i q u e

B

study of ma tr ice s.

We b e g i n w i t h the

element

an d

of the R in gs

reference

Definition

y

we hav e

i s o m o r p h i s m on the u n d e r l y i n g

section presents in our

= A • y(v) = T b o

and

.

be u s e d f o r r e f e r e n c e specific

• v)

.

.

Si n ce

4.

v e MA

for all

yis an R [ A ] - i s o m o r p h i s m

y o T a (v ) = y (A

for all

• v

= 0

impli es

r

of a ring a = 0

is a c o m m u t a t i v e

in

R

Q

is an

.

ring w i t h at

the total q u o t i e n t r i n g

satisfy in g:

R

of

least R

is

20

i)

Q

has

ii)

R

is a su br in g of

iii)

an identity,

Every regular in ver se

iv)

in

b

Sin ce

a

the

clear

that w h e n e v e r

of

R

has

of

R

of

Q

b

is of the

fo r m

are el em e n ts

of

ab” 1 R

and

element.

Q

of

R

is an int egral

.

an

,

and

ring

f ie ld

R

has

an identity,

is w e l l - d e f i n e d .

We have

domain,

the f o l l o w i n g

the

It is Q

is the

result

for a

sum of rings.

T h e o r e m 1.6 with

of

ri n g we are c o n s i d e r i n g

quotient

direct

Q

is a re gu l a r

total

quotient

e l em en t

E v e r y el em en t where

Q ,

Suppose

i d en ti ty S^

for

the to tal Proof

1

S j , S 2 , . . . ,Sk

and

that

i=l,2,...,k.

quotient

ring of

We fi rst n o t e that

1 ©

...

© 1

the

same

Qi

and

that

identity.

S

are c o m m u t a t i v e rings

is the total q u o t i e n t Then

Q = Q 1 @ . . . @ Q k

S = Sx © S

...

© Sk

is a ring w i t h

is a su bri ng of

The r e g u l a r

ele me nt s

Q of

ring is

. i de nt it y , w h i c h has S

are of the

fo r m

a = a1 ©

...

where

a^^

is a r e g u l a r e l e m e n t of

a f 1 ©

...

© a k _1

of

Q

.

is the

© ak

in ver se of

Sj^ . a

Thus and

a -1 =

is an element

21

Let

x = Xj

Qi

is the

b^

is r e g u l a r

total

x = a j b i -1 ©

Since 1.4,

Q

identity i)

quotient

...

Q

We n o w

. . . © Xj,

in

be any e l e m e n t

ring of

, x^

, i = l,2,...,k

© a kb k _1

satisfies

is the

Definition

©

total

=

(ax ©

all

.

...

ring

© a k ) Cb x ©

of

1.5

[2]

1 , and

Let R

S

s e S

if

s

S

R iii)

, S

The

closure S iv)

with

[]

of

R

de pe ndence.

ring w i t h

that

1 e R

.

S

is

in

of

ove r

S

is c a l l e d S an d

R

in tegral

in te gr al

elements R

in

. over R

.

that are

the i n t e g r a l

is a s u b r i n g of

.

If

R

we

say

Throughout monic

over

© b k ) -1

is the root of a m o n i c

is said to be

int e g r a l

...

is said to be i n t e g r a l

e l e m e n t of

set of all

wh er e

.

such

polynomial with coefficients If e v e r y

S in ce

of D e f i n i t i o n

be a c o m m u t a t i v e

a su br i n g of

An element R

S

.

Th er e f o r e ,

the c o n d i t i o n s

quotient

Q

= a ^ b ^ -1

look at the p r o p e r t y of in tegral

ov e r

ii)

of

polynomial

is R

its own

in te gr al

is i n t e g r a l l y

the rest of this in

identity whose

R[A]

, where

quotient

c l os u r e

closed

section, R

in

let

in

S

S

,

.

f(A)

be a

is a c o m m u t a t i v e

ring w e d e n o t e by

Q

.

ring

Set

.

22

A = R[A] / (f (A.))

N o t e that identity. el e m e n t s of

in t e g r a l

indeterminate

i n te gr al

ov e r

i n te gr al

ov er

Theorem

1.7

0

R R

Let

is the

R

is c l e a r l y

e l e m e n t of

R

A

A

t i v i t y of

and

, having

The

A

be the

in t e g r a l

a s u b r i n g of is

i nt eg ra l

dependence

is,

g(A)

e 0

f(A)

, where

according denotes the

as

f o rm al

^ 0

are

A

.

Let A

0

is

in

A

in t e g r a l

ove r

be any

By the

g(A)

transi­

is

is the

in te g r a l

int egr al

.

[]

is s e pa ra bi li ty .

We

domain with quotient

irreducible

or

Then

polynomial

is se p a r a b l e g'(A)

d e r i v a t i v e of

A =

.

g(A)£

or i n s e p a r a b l e

= 0 , where

g(A)

.

.

g(A) .

R

A

d i m e n s i o n a l Q-algebra.

is a field,

g'(A)

in

be a m o n i c po ly n o m i a l .

[32] A n K

of

w h i c h is

Thus

is an in t e g r a l

isa fin it e

1.6

A

ove r

di s c u s s

R

0

A

field

Definition

.

of

and pow er s

, we see that

[2, p 60],

in

R

Q l>]/ (f O ) )

0

e v e r y e l eme nt R

cl os ur e

c l o s u r e of

is a s u b r i n g of

Let

A

0

final p r o p e r t y we

.

= 0 in further,

n o w a s s u m e that Q

f(A)

the same

of in te g r a l

of

ele me nt s

We ha v e

which

, that

Si nce

.

no t e that

.

integral

c l o s u r e of

K[A]

A

and p r o d u c t s

R

A .

int e g ra l

Since

, A

of

sums

over

and

Proof

over

that

A= Q[ A ]/ (f (A ))

is a sum of p r o d u c t s of e l em en ts

of the

Then

is a s u b r i n g

We re ca l l are

A

A

an d

g'(A)

23

Using

standard references

it c a n be s h o w n that only

if

f(A)

is

A

is a s e p a r a b l e

the p r o d u c t

separable polynomials. for

A

(for ex amp le

Q-algebra

o f disti nct ,

Th i s w i l l

serve

[6 ] or

[32]), if and

ir re du c ib le ,

as our c r i t e r i o n

to be se parable.

Suppose

f(A)

f(A)

into di s t i n c t ,

Since

the

ideals

k II f^CA) i= 1

=

is

irreducible,

(f^CA))

relatively co-maximal,

and

we m a y

the

f a c t o r i z a t i o n of

separable polynomials.

(fj(A)) apply

, for

i f j

the C h i n e s e

, are

Remainder

Theorem and write

A = A\

where

^

= Q[A] / (f^ (A) )

extension of A

Q

is s e m i - s i m p l e .

a polynomial In this be

.

A

A

is s e p a r a b l e

this

Suppose

and

Let

if

be

important

is a D e d e k i n d domain,

Q

see

Q

is

that zero,

f(A)

ca n

the d i s c u s s i o n of a

integrally closed domain

is m a x i m a l . f(A)

if a n d on l y

section with

ch a p t e r s .

R

of

we

irreducible polynomials.

situation which will

ideal

is s ep ar ab le ,

algebraic

if a n d o n l y if it is i r re d u c i b l e .

particular

noetherian,

© i4k

If the c h a r a c t e r i s t i c

into d i s t i n c t

We c o n c l u d e

...

is a f i n i t e s e p a r a b l e if

is s e p a r a b l e

case,

factored

Thus,

©

in w h i c h

be the q u o t i e n t

a monic polynomial.

Suppose

in later that

is,

every prime

fi el d o f

that

a

f(A)

R is

24

the p r o d u c t mial s,

of di s t i nc t,

so that

As

ir red uci ble ,

A = Q [ A] / ( f ( A ) )

above,

i . We

is a finite, Let

de no t e

then ha v e

[4, V

the

§1]

in t e g r a l

We ha v e f i e l d of

0 j_

for all

the t o t al

quotient

© Ak

e x t e n s i o n of

int egr al

cl os ur e

of

Q R

, for all in

^

.

that

c l o s u r e of

further

...

separable

0 = 0x ©

is the

is a s e p a r a b l e Q -a lge bra .

we write

A = Ax ©

where

separable p o l y n o ­

[31,

R

in

p 141]

i .

ri n g of

...

© (?k

A that

. A^ is the q u o t i e n t

Thus, by T h e o r e m 1.6, 0

.

A

is

Chapter Lattices

1.

Definitions

and B a si c

Throughout with

this

the q u o t i e n t

over Ord er s

Re sul ts

chapter,

f ie ld the

Q

R

.

\^e ha v e

Definition

2.1

A n R -ord er

S

containing

the

i d e n t i t y of

A

S

is a f i n i t e l y

ii)

S

contains

S-module which considered

We

in

A is a s u b r i n g of

a Q-basis

A ,

, such that

g e n e r a t e d R- module;

A n S -l at ti ce

of

A

, i.e.

of an R - o r d e r

and QS = A

S

is an

is f i n i t e l y g e n e r a t e d and t o r s i o n - f r e e w h e n

say that

S

is a m a x i m a l

contained

is g i v e n

every R-order

S

in

R-order

in any o th er

a s e p a r a b l e Q- al g e b r a ,

An e x a m p l e

a finitely generated

as a n R-module.

not properly is n o t

A

f o l l o w i n g de fin iti ons .

i)

2.2

will be a D e d e k i n d d o ma in

For

Q- a l g e b r a ,

Definition

2

R-order

maximal

[25, p

201].

is c o n t a i n e d

orders If

A

in a m a x i m a l

is a c o m m u t a t i v e ,

separable Q-algebra,

maximal

R-order.

Maximal

special

pr operty.

R-orders

25

in

have

there the

A in

if A

.

n e e d not is

S

is

If

A

exist.

separ ab le ,

R-order.

If

A

is e x a c t l y one following

26

Theorem

2.1

[24, p 178]

Suppose

algebra and

S

S-lattice

S - pr oj e c ti ve .

is

We n o w A

by

.

a pp ly

M

Set

these of

Rn

the A - m o d u l e A

that

see

A

M

is a A - m o d u l e

and

is an R - o r d e r

in

A = Q [ A ]/ (f (A ))

A

.

A

S u p p o s e n ow that

Let

R

A

.

and

QA = A

,

By its co n s t r u c t i o n ,

Thus,

M

free,

of d i s t i n c t

h en ce

is a A-latti ce.

is a s e p a r a b l e Q-a lg eb ra .

is the p r o d u c t

D e no te

.

is f i n i t e l y g e n e r a t e d a nd

t o r s i o n - f r e e , as an R-mod ule .

f(A)

Then.every

with characteristic polynomial

a s s o c i a t e d to a m a t r i x

which

Q-

to our situat ion .

is f i n i t e l y g e n e r a t e d over

we

is,

in A .

definitions

A = R[A]/(f(A))

Since

is a s e p a r a b l e

is a m a x i m a l R - o r d e r

be an el em e n t

f(A)

A

irreducible

Th at

se p a r a b l e

k polynomials,

f(A)

=

II

f^CA) , and we m ay xvrite

A =

i= 1 Ax ©

...

we ha v e

©

where

the

[25, p 179]

Q-algebra

Sis an R - o r d e r i)

S

in

A

A

Let

S

In this

be a su bri ng

, h a v i n g the same

situation

of the

identity

as

A

.

if and onl y if

is a left R- mo dul e;

ii)

S

co n t a i n s a Q - b a s i s

iii) S

is

int eg ra l

Since

A

is

satisfies

.

f o l l o w i n g c h a r a c t e r i z a t i o n of R-orders.

T h e o r e m 2.2 separable

= Q [ A ] / ( f i (A))

over

int eg ra l

the a bo ve

R

ov er

con di ti on s.

of

A , i.e.

QS = A

;

.

R

, it is c l e a r

Let

0

be the

that

A

int egral

27

c l o su re

of

R-order

in

R A

R

S

containing in

If

A

this

i

for

@ A

and

.

.

Then

Th at is,

we

first

i f j 0^

0

in

set

(fj_(A))

of

if A^

= 0

, A = 0

and

A = Aj

©

...

c l o s u r e of R

in

A

is

R

R-order

for

and u s e d to d e t e r m i n e

can be c o m p u t e d in W e i s s

section,

© Ak

.

If

in

2^

,

in

A

the d i s c r i m i n a n t s

, the

... , . When

of

A^

eq ua li ty , as

Condition

we wil l

be d i s c u s s i n g a s i t u a t i o n

the n u m b e r of i s o m o r p h i s m c l a s s e s of m o d u l e s

is

are

[31].

The J o r d a n - Z a s s e n h a u s

a t e d to m a t r i c e s ,

by

, i =

i = l,2,...,k

0^

Q

A

determine

0 = 01 ®

and

on

in

Then,

(fj(A))

field,

of m a t r i c e s ,

i nt eg ra l

strictly

.

To

is an a l g e b r a i c n u m b e r

which

is

Aj_ = R[ A] /( f^ (A))

in t e g r a l

closure

A

A-module.

Hen ce , Ais the m a x i m a l

In this

S

inA

Q

2.

is an

is the u n i q u e m a x i m a l

, we h a v e

the

= 0 , if and o n l y

discussed

0

e ve ry R - o r d e r

there is no R - o r d e r

is a p r o j e c t i v e

i n te g ra l

[4].

0

If the ideals

we d e n o t e by

co mm ut a ti ve ,

in A

integrally closed

occurs,

co m a x i m a l

S

Therefore,

M

.

is

.

is

l,2,...,k

then the

0

.

2.1,

0

is any R - o r d e r

in

0

R-order

when

S i nc e

and we have

is c o n t a i n e d

Theorem

.

.

Suppose over

in A

hence

finite.

quotient

in

associ­

the n u m b e r of s i m i l a r i t y c l a s s e s We w i l l

fi eld of

R

need a further .

assumption

28 Definition i)

2.5 K

Let

be a field.

is an a l g e b r a i c

fi nit e ii)

K

K

number field

e x t e n s i o n of the

is a f u n c t i o n f i e l d

rational

if it is

e x t e n s i o n of a f i e l d of ra ti ona l k(X) iii)

K

, where

is said

eit he r

k

if it

is a

f ie ld

Q

.

a fini te f u nct io ns,

is a fin it e field,

to be a g l o b a l

field

an a l g e b r a i c n u m b e r

if it

is

field or a

f u n c t i o n field.

When will

Q

is a glo b al

be useful.

ideal

classes.

this

Dedekind domain whose

s e c t i o n we quotient

c h a r a c t e r i s t i c po ly n o m i a l .

will in

Rn

d en o t e

.

Set

We a s s u m e A

has p r o p e r t i e s

R

for any

be c o n s i d e r i n g m a t r i c e s

A = Q[ A] / ( f ( A ) )

R

In p a r t i c u l a r , S ec o n dl y,

Throughout

We will

field,

has a finite n u m b e r a e R , R/aR

as su m e that

field of

that

Q Rn

is

R

finite.

is

a

is a g l o b a l ha v i n g

f(A)

A = R[ A] /(f(A)) is a se p a r ab le

the A - l a t t i c e a s s o c i a t e d to

of

field. as

and Q-a lg eb ra .

the m a t r i x

A

.

For

any R - o r d e r

S

in

A

may construct

the A - m o d u l e

QM = Q 0 M R

.

To

A = Q ® S R

from the d e f i n i t i o n of R-order,

Q ® M = R

see

Q ©

that

QM

and any S - l a t t i c e

QM

in the ca n o n i c a l is an A - m od ul e,

(S 0 M) = (Q ® S) ® M = R S R S

M

, we

way:

we n o t e

and we

A 0 M S

have

.

that

29

We t h e r e f o r e

identify

M

consider

embedded

in

M

By c o n s t r u c t i o n , {ei,...,en } •

ra n k of

of

T h e r a n k of

Q

to m a t r i c e s

of A - l a t t i c e s

Our d e v e l o p m e n t n e e d the c o n c e p t

Definition

2.4

n

.

of

Rn

are all

an R - b a s i s

corresponds

and

there

is

.

n

Als o,

, the

the A - l a t t i c e s

of R - r a n k

n

.

to m a t r i c e s [23].

is finite. We

first

be any R - o r d e r

in A

= A

that of

.

A n S -ideal

.

are n o t n e c e s ­

this d e f i n i t i o n e x c l u d e s

r < n

elements.

as R - m o d u l e s ,

zero d i v i s o r s ,

ide al s

S u c h ideals

si nc e they c o n s i s t so they are n o t

of

A-lattices.

algebraic number

th eo r y w h i c h

excludes

ideal.

Definition ol

R

to the r e s t r i c t i o n of the d e f i n i t i o n of

in c l a s s i c a l

zero

MA

a s s o c i a t e d to m a t r i c e s

Note

e l e m e n t s w h i c h are

the

and

the n u m b e r of A - i s o m o r p h i s m

suc h that Q

are not torsion-free

ideal

in

that of R e i n e r

S

A Z ,...,A^

that

Sji

, it

is a fin it e d i m e n s i o n a l

dimensional

n u m b e r of

in the

is a glob al

A = Ax @

are R - o r d e r s

S

el em en t

Ho we ve r,

Q -a lg e b r a ,

Proof

ring of

a non-zero

is i r r e d u c i b l e

2.3

number

.

r e m a r k that the class n u m b e r

i nt e g r a l

the c l a s s

.

not n e c e s s a r i l y

of

is ca l le d

h(S)

and

exists

is

Lem ma

of S - i d e a l s

, and

fo rm in g

ideals,

one

of each

is

ideal

all p o s s i b l e

from ea ch

The n u m b e r of S - i d e a l

S^

,

cl as se s k

d e t e r m i n e d by this

set

is at m o s t

the p r o d u c t

II

hCSj^)

,

the re

is

i= 1 which

is finite.

a decomposition i

Further , Oi* -

is an S^-ideal.

if ©

Hence

...

c/l,

©

is any S -i de al , [32, p 175],

is in the same

wh ere

cl as s

as

31

one of the

idea ls

c o n s t r u c t e d a bo ve and

h(S)

is

finite.

[]

T h e o r e m 2.4 the n

[23]

h(0) S in ce

where

A^

in

R-order

in

,

A

A

is separa bl e,

...

© 0k

we w r i t e

is a glob al

where

0^

A = A1 ©

field.

... © A^.

Therefore

is the i n t e g r a l

closure

of

R

Aj_ .

A^

0i

is a D e d e k i n d d o m a i n w i t h q u o t i e n t

, a gl ob a l

for all

i

field,

.

so the

Therefore,

class n u m b e r of

by L e m m a

2.3,

0^

field is finite

the cl as s n u m b e r of

is finite.

0

Theorem .

2.5

If two

[]

[23]

Let

O-ideals

in the

same O - i d e a l

Proof

Suppose

are

A

.

1 .6 ],

S in ce

A

Definition

2.6

Zassenhaus

condition,

are

[23,

p

are

isomor­

to an A-

be g i v e n by m u l t i p l i c a t i o n by a unit

in the same

224]

We

JZ(S)

m a n y S - i s o m o r p h i s m cl ass es .

in

t h e n they are

are 0 - ideals w h i c h

for e v e r y p o s i t i v e

t

R-order

isomorphic 0-latices,

is the total q u o t i e n t rin g

and

mo s t

the m a x i m a l

The 0 - i s o m o r p h i s m exten ds

Cns

if,

denote

class.

i s o m o r p h i s m w h i c h mu s t of

0

and

p h i c 0 - l a t t i c e s.

A

is the m a x i m a l

= Q[A]/(f^(A))

Each

A

0

is finite.

Proof

0 = 01 ©

If

ideal

say that

holds

integ er

of

0

class.

[]

the Jo v d a n -

for an R - o r d e r t

, there

of S - l a t t i c e s

[Theorem

exist

having

S

in

fi ni te ly

R - r a n k at

32

We wi l l

sh ow that

field.

S in ce

l at ic e s

of R - r a n k

similarity

Theorem Proof tive

the m o d u l e s

for

Q

a s s o c i a t e d to m a t r i c e s

, this

implies

the m a x i m a l

R-order

Since

0

is m a x i m a l,

every O-lattice

choices

isomorphic M

How ever,

is om or ph is m,

is bo unded,

i s o m o r p h i s m cl ass es

For

Let

t

2.4,

h(0 )

T h e r ef or e,

, JZ(O) M

are

Up

only

f in i t e l y

[]

A = R [ A ]/ (f (A ))

M

A

, JZ(A)

in t e g e r

set of r e p r e s e n t a t i v e s

in

holds.

and let

{L j j . - . j L p }

of 0 - i s o m o r p h i s m cl ass es t

, where

0

embedded

0M

0M = M

Lj

is the

.

M

Since

to

of b o u n d e d R-rank,

Considering

OM =

is

of p o s s i b l e

any A - l a t t i c e of R - r a n k at mo s t

replacing

is p r o j e c ­

is finite.

be

.

holds.

sum of O- ideals.

there

of O - l a t t i c e s

be any p o s i t i v e

R-order

Let

A

the nu m b e r of su mm a n d s

of 0 -la tt ic es w i t h R - r a n k at mo s t maximal

of

holds.

2.6

be a full

in

are o n l y a fini te n u m b e r

for each summand.

Theorem

0

to a dire ct

by T h e o r e m

there

JZ(O)

are A-

a finite n u m b e r

For

of

a global

of ma tr ic es .

and t h e r e f o r e

Proof

holds

2.5

bounded.

i.e.

n

c la ss es

If the R - r a n k

many

JS(A)

in

QM

for some

by a A - i s o m o r p h i c

t

.

, we can form the 0 - l a t t i c e j

,1£

co py

j _< p

, we m a y assume,

if n ec es sa ry ,

that

Lj . Choosing a non-zero

we have

el em e n t

a e R

such that

aO c A

,

33

aLj = aOM Since

Lj

is an O- l a t t i ce ,

R-module.

Since

therefore

Lj/aLj

Q

an R / a R - s u b m o d u l e nu m b e r was

ar bit ra r y,

Proof

2.7

S i n ce

Lj/aLj

of

ible

separable

ity cl ass es nomial Proof

3.

f(A) This

choic es

The

c lass n u m b e r of

factors,

of m a t r i c e s

fi n it e and

M/aLj

is

M

.The

choice

we

[]

A

is finite.

can u s e

2.5 to show

an a r g u m e n t

h(A)

is the p r o d u c t there in

M

of

holds.

holds,

f (A )

is

Ho we ve r,

for

JZ (A)

If

R/aR

, so t he re are on l y a finite

so

JZ(A)

2.8

field,

is a fin it e group.

to the p r o o f of T h e o r e m

Corollary

oM = Lj .

it is a f i n i t e l y g e n e r a t e d

is a global

of n o n - i s o m o r p h i c

Corollary

c AM c M c

si mi la r

finite.

[]

of d i s t i n c t

irreduc­

is a f i n i t e n u m b e r of s i m i l a r ­

Rn

having

characteristic p o l y ­

. f o l lo ws

d i r e c t l y from T h e o r e m s

1.5 an d

2.6.

G en e r a of L a t t i c e s Throughout

quotient f(A)

field

is a m o n i c

this Q

section,

polynomial Set

0

of

l o c a l i z a t i o n of

closure R

The d e v e l o p m e n t genus,

is a D e d e k i n d d o m a i n w h o s e

is an a l g e b r a i c n u m b e r

a s e p a r a b l e Q- al g e b r a . the in te g ra l

R

such that

field.

A =Q [ A ] / ( f ( A ) )

A = R[A]/(f(A)) R

in

at the p r i m e

A

We

.

ideal

Rp p

as s u m e is

a n d d e n o te will

denote

follo ws

the

.

of the c l a s s i f i c a t i o n of A - l a t t i c e s

w h i c h we p r e s e n t here,

by

Roggenkamp

by

[26, VII].

34 Let As

S

be an R - o r d e r

in the p r e v i o u s

Set

Sp = Rp ® S R

Rp-order

and

Definition in the

2.7

section,

a nd

Mp

Two

We

concepts

1.2]. p

Note

Definition of

R

and

r a t he r

coincide also

, it fo ll ow s

.

is an .4-module.

Not e

Mp

su c h that

and

ideals

that

N , are

Sp

Np

are

p of

is an

R

Mp = Np QN

S denote

A p is not

a nd QN

are

.

in terms of pH o we ve r,

as s ho wn

for

to be

isomorphic

than lo ca li za ti on s.

if

said

QM

is o f t e n d e f i n e d

Q M a nd

Let

M and

for S- l a t t i c es ,

that

that

2.8

an S- lat tic e.

, if and on l y if

that genus

adic completions two

M

QM = Q ® M R

= Rp ® M R

M V N

all p r i m e

remark

and

S- la t t i c e s ,

isomorphic A-modules for

Mp

A

is an Sp-latt ic e.

same gen us,

Sp-modules

in

in

these

[26, VI,

even one p r i m e

ideal

are A - i s o m o r p h i c .

the set of p r i m e

a maximal

c a l l e d the set o f c r i t i c a l p r i m e s

of

ideals

Rp-order. R

and

S

p is

is a fin it e

set.

In o t h e r word s, that

Ap ^ Op

Ap = Op

.

For

, w e hav e

A-lattices

M

S

the p r i m e

that

an d

d e f i n i t i o n of genus:

is the set

N

ideals

Q M = QN [11]. M V N

of p r i m e p

implies

Thus we

idea ls

such that M p = Np

,

can r e p h r a s e

if a nd on l y

p

if

QM

for the

a nd

QN

such

35

are A - i s o m o r p h i c p e S

and

and

Np

are A p - i s o m o r p h i c

for all

. We

see that

a genus

A - i s o m o r p h i s m cl as se s Jordan-Zassenhaus 4.5,

Mp

contains

a finite n u m b e r

from the f o l l o w i n g

Theorem.

of

statement

A p r o o f m a y be

found

of the

in

[26, VI,

4.7].

Theorem

2.9

algebra

A

Let .

S

be any R - o r d e r

For W

any

in the s e p a r a b l e Q-

f i n i t e l y g e n e r a t e d A - m o du le ,

the

set

o(W)

contains

= {M |M

on l y a fin it e n u m b e r

It is c l e a r th at

M V N

o(QM)

is an S - l a t t i c e

that

, th en

if

Theorem

2.10

and

using

f o u n d in [24,

Let W

QM = W}

of n o n - i s o m o r p h i c

N e a(QM)

c an be co mp ute d,

p r o o f can be

M

and

.

N

are A - l a t t i c e s

The n u m b e r

the

S- l a t t i ce s.

following

such

of g e n e r a

in

theorem.

Its

173-175].

be a f i n i t e l y g e n e r a t e d A - m o d u l e

and

set

a(W)

= {M| M

is a A - l a t t i c e

Th e n u m b e r

of g e n e r a

where

is

c(W)

hp

in

cr(W)

suc h that

is g i v e n by

Q M = W)

g

=

II hp p eS

the n u m b e r of A p - i s o m o r p h i s m cl as s e s

for

eac h

p e S

Corollary

2.11

Suppose

.

in

.

A = 0

, the m a x i m a l

R-order

in A

.

36

Let Rn

MA

be

the

A-lattice

associated

with characteristic polynomial

o n l y one ge nu s Proof

Since

for all

p

in A

.

f(A)

.

Th e n

A

in

there

is

o(QMA ) . is ma xi m a l,

Henc e,

any A -l at ti ce ,

to the m a t r i x

Ap

is a ma xi m a l

by our p r e v i o u s

N V M

if and on l y

remarks,

if

Rp - o r d e r if

QN = Q M A

N

is

as

d-m od u l e s .

A

fi ne r c l a s s i f i c a t i o n of A - l a t t i c e s

to co un t present apply

[]

the n u m b e r

of i s o m o r p h i s m c la ss es

the d e f i n i t i o n

it

to our

will

enabl e us

in a genus.

for an a r b i t r a r y R - o r d e r

specific

s i t u a t i o n of lat tic es

We

and then a s s o c i a t e d to

matrices.

Definition

2.9

Ti\ro S - l a t t i c e s

in the same r e s t r i c t e d genus, following conditions i)

M

ii)

and

OM

and

A

N

a t e d to

A

.

Set

g (M ) = ( A - l a t t i c e s components al g e b r a s , [11],

of we

ON

are

V N

are

said to be

, if and o n l y if the

iso mo rp hi c,

R-order

in

and de no t e

where

containing

Rn

having by

MA

N|N

^ MA) .

ca n a p p l y

the

are

S

is

as c h a r a c ­

the A - l a t t i c e a s s o c i ­ N|N

V M}

and

S i n c e none of the simple totally definite

f o l l o w i n g result s

in R o g g e n k a m p

0 .

f(A)

0(MA ) = (A-lattices

H o m A (MA ,MB)

as g i v e n

V

M

N

are in the same genus;

be a m a t r i x

teristic polynomial

and

hold.

the m a x i m a l

Let

M

[26, VIII].

quaternion

of J a c o b i n s k i

37 T h e o r e m 2.12

[26,

p 103]

r e s t r i c t e d genera. we can

f 0 .

invariant

Ma

invariant

method

...

Also,

if

G 1 = GS

,

is a unit

d e f i n e d by

find a c o r r e s p o n d i n g

Ag a i n ,

the class

of the

1.5, we

have that

i s o m o r p h i s m class

ideal m a t r i x

of the of

the

ideal

A

ideal

. class

of the m o d u l e

. We

We

In

n c K e -j) ~ 1 g'-.u. i= 1 13 3

,

of the s i m i l a r i t y class

Recalling Theorem is an

G'

.

is om o r p h i s m .

de t( g j j ) an

ideal.

det(GS)

is an

R

m a y be r e p l a c e d by

d e t e r m i n e d by

n 1 Siju j i= 1

=



1 .

=

the

(A - a)g(A) fo rm

f(A)

In that case,

and =

we

refer

s i m i l a r i t y co nd it ion s.

= g(A)h(A)

section

and

the m a t r i c e s

h(A) A

and

has

degree

A r

hav e

2 . bl oc k

and

=

A’ =

(a'jj)

Ai

A.*

1

and A ’j

are

polynomial Ai*

and A \

in R n - 2 g(A)

are

polynomial

h(A)

and A ' 2

are

the

zero m a t r i c e s are A

and

an N>

i j )

o

( a

CM