GENERALIZED EQUIVALENCE OF MATRICES OVER PREFER DOMAINS

Internat. J. Math. & Math. Scl.

VOL. 14 NO. 4 (1991) 665-674

665

GENERALIZED EQUIVALENCE OF MATRICES OVER PREFER DOMAINS

FRANK DEMEYERandHAINYAKAKAKHAIL

Department

of Mathenatics

Colorado

State

University

Fort

Collins,

CO U S A

80523

21A Victoria Park The

Mall,

Lahore Pakistan

(Received April 19, 1990)

ABSTRACT: Two

mx n matrices A,B over a commutative ring Rareequivalent

i,.ve,-tible nmtrices P,

O

^{over R withB} PAQ. While anymxnmatrixovera principleideal ^dota.i,

ca, bediagonalized, thesame is not true for Dedekind domains. The first author and

T..I.

Ford ittroduced acoarserequivalencerelationonmatricescalledhomotopyandshowed any x mtrix over a. Dedekind domain is homotopic to a direct stun of x2 matrices.

In

^this article wc giw, necessary and sufficient conditions on a Prefer domain that any mx n matrix be

homotolfic

^{to a.}

directsumof x 2matrices.

l(cy Words andPhrases: Priifer^domain, Progenerator module, Bezoutdomain,matrixequivalence 1980 Subject classification codes 13F05, 13C10, 15A33.

1.

IN’I’RODUCTION

Let M,N be finitely generated projective faithful modules

(progenerators)

over a. co,nmuta.tivc rig R.

An

R homomorphism h M N is called image split in ^case h(M) is ^a faithful R-direct summand of N. If

I

^{:M N and g:P Q are} homomorphisms of R progenerators ^then

are said to behomotopic ifthereare image split homomorphisms ^h A B and isomorphisms

,

makingthecommuting diagramofR-modules

MA I

N@B

PC

O

^D

If

I ^P-’g

for isomorphisms p,v then

I

^{and g are homotopic}

(where

R A B C O h. ln). Thus equivalent homomorphisms ^are homotopic but not conversely. The notion of homotopyofhomomorphismswintroduced in

[4]

^{toremovemost ofthe}obstructionobserved by

L.

Levy

in

[13]

^todiagonalizationof matrix transformationsunder equivalenceoverDedekind domains.

Summarizingsomeof the results in

[4],

homotopyis anequivalence relationonhomomorphisms of progenerator modules and tensorproduct ofhomomorphis inducamultiplicationon homotopy classeswhich turns this setof cls intoamonoiddenoted M(R). Each

homotopy

classisreprescnted by at let onematrix transformation, and if R is a Dedekind domMn

by

a matrix transforma.tio which isadirectsumof x 2matrices,amatrixof the form

a b 0 0

0 0 a. be

m x2m

666 ^F. DeMEYER AND H. KAKAKHAIL

Moreover, if Ij ajR

+ bR

^then

^I

^{D 1._, D D IfR is a} discrete valuation ritg lwn .(

PN[a.]. the mouoid of primitive polynofia,ls with coefficients iu N

{0,1,2,..} togctltcr

^wil,l 0-1)olytomial. If^{R is a}Dedekind domai tlen M(R) isnaturally isomorphic^to

(],,,w,st,m

⁽

atttl tltis isomorphism gives an

isomorl)hisn

between M(R)

and

primitive

polynomials

^{over N} ideterminates indexed byMazSpec(R).

q’le purpose of this paperis todetermine the extent to which theseresults can begctwralizcd toarbitrary doma.ins.

In fact,

theycomeclosetocharacterizingDedekind domains. Wefirst olsct tltat ifli’ isacommutativering containing^a maximal ideal Psuch that dinn/t.( P/

P")

>2tlt(;t, isahonotopy class in M(R) which contains ^nomatrix transformation which is ^adirect^sutnof x

tmtriccs.

Thus,

if R is a Noetherian domain andevery

homotopy

class in 3A(R) contains ^{a ml,}rix wlich isa direct sumof x2 matrices then direr

<

1. The inclusion mapfrom a domain R to its itcgralclosure

R

inducesamonoidhomomorphism.1(/)-

(/)

^whichwassteadied in

[6].

^llcrc

relax ^l,]eNoetherian condition andstudy.(/) for Prfifer domains. IfRisaPriifer^doma,io1"Krll ditnension orif Ris a Priiferdomain of finite character

(each

nonzero element,of Ris

i otly finitely many maximal ideals) ^weshow every class in .A(R) ^contains a represet|ting natrix wlicl is adirectsumof x2matrices. IfR isany valuation domain with valuegt’oup^(:;<(R, +) (;+ is 1,]e nonoidofnonnegativeelements ofGformthemonoid PN(G+)^of"primitive l)olynomials"

:.,,,:+ o.x.

^with

^a

^N,ahnostall

a

⁰and 9cd{%lgeG^{+ 1.}

We

showj4(R)

PN(G+).

After givi,g

asliglt generalizationof

L. Levy’s

"Separated DivisorTheorem" formatricesoverDedekittd donains

[13],

^wecan show for Priifer domains that .I(R) ^is naturally isomorphic ^to

t.,nt.sr.(n).,’vl(Rt,)

^it"

and o.ly if R isof finite characterand the valuation ringsat the maximal ideals of Rarc pairwisc independent. The principal examples ofPrfifer domainsoffinitecharacter whosevaltation rings at

m,ximal idealsarepairwise independentareDedekind domains and valuation donmins.

Partof this paper appeared in the first author’s Ph.D. dissertation written at Colorado Uniw:rsity. This paper was

completed

while the second author was a visitor at Florida Atlantic University. Hewishes to thank departmentchairman Jim

Brewer

for hishospitality. We would also like to thank

L. Levy

for hishelp withthe proofofthe generalized

Separated

DivisorTheorem.

"2.

SEPARATED DIVISOR THEOREM

PROPOSITION

1.

Let

R be a conunutative ring containirg ^a maxinal ideal ^p wil.l dimn/t,(P/P

’)

> 1. Then there is a matrix transformationover R which is not homotopic to a di- rect sumof x9.matrixtransformations.

PROOF: Let .

^{R Sbea}homomorphismof commutativerings,^so Sisan

R-algebra.

Then itducesamonoidhomomorphismM(q)’M(R)-M(S) byM()(lf[)= [I(R)Yl, where^if

y

Homn(N,U.) the 13" Hotns(SC)U,,S(R)U,.)

(Theorem

^of

[4]).

^Sinceeach class in

M(S)is represented

byamatrix transformation, if0^isanepimorphism then M() isanepimorphism. If isanepimorphism and ^if every class in M(R)isrepresented byamatrix whichisadirectsumof 2matrices,then ew.ry class i M(S) is

represented by

amatrix which is^adirectsumof x :2matrices.

Thus,

it suffices to check tle conclusionof the propositionforahomomorphic image ofR.

Let

{ax

+ ^P,a+ P-} u

{c,,

+

P"},t ^be

a basis for

P/P"

^over

RIP.

^LetJbe the ideal^in/

generated

by

P

and {a,},,t. The ringS R/J ^isa local ring with maximal ideal M P

+

J/J.

Moreover,

M (0) and

dinslM(M/M)

^2.

Let

a,a.., M be linearly independent over S]M

We

check the matrix aa

a:/

^is not homotopicover S toally

0 ax

GENERALIZED EQUIVALENCE OF MATRICES OVER PRUFER DOMAINS 667

i.at,’ixof the form

H ⁰

..,

SinceS islocal,with respecttoasuitable basischoice, eachimagesplit homomorphismhasa ,natrix

representation ofthe formdiag(l,..,1,0,..,0) (Proposition

3(9)

^of

[4]). ^We

^needtocheck

F=

aO aa’2]

^{diag(1 1,0, ,0)=}

mx2m

is ,lot equivalent over Sto H. View Fas the relation matrix of thefactor module

SO")/Lr

where l,r is thesubmodule ofS^(2")

generated

by the rows ofF. Then

S(2)/Lr

^{is isomorphic to adirect}

sul, of modules of the form A S

S

< (a,a2),(O, al) >

together

with

0-summands. A

^(lit’oct

(’alcttlation shows the S-endomorphisms ^{of S,S}

leaving

< (a,a2),(O,a) > invariant ^are givett

"’atric"s t’the frn

aft a+m’n’

^’vhere

^{’n’’’’’ M’’}

^{S" That}

^is’

^{Ends(A) is a}

hmm"l)"ic

imgo of the ring of these natrices.

A

direct but msy calculation shows that if

+,nJ

[ +mJ

^{m’ then e=O, nd fl=,, =m’=O-} Sinceidempotents^can belifted modulo nilpotent e,B S,m m’ M

o

Ends(A) has kernel ideal and the natural homomorphism from

e

+

^m

+

^{,,m,m’ whichisnilpotentwes}Ends(A) hasnoidempotentsother than 0and

so

A

isan indecomposable S-module.

In

thesame way^{view Gas}the relation matrix of the factor ,odue

S"’/L;

where

La

^isthe submoduleofS

generated by

the rowsof^G. Then

S’"I/La

^is

ismnorphic toa.direct sumofmodulesoftheform B, S S/<(a,,B,)^>.

An

easycalculatonshows d,sh(M"A)

, dmsm(M.

^B,)

a,

and dimsl(A/MA) ² dimsl(B,/MB,)(1 m). IfA thenA/MA/ML B,/MB,^soL MLsoby Nekaya’slemma L (0).

In

thisceA B, which is impossibleby thefirst dimensioncountabove. Thus

A

isan indecomposable S-modulewhich is not adirectsummand ofany B,.

By

the Krull-Schdt Threm

Somalia ^S(/

^and F,G cannot be equivalent matricesoverS.

REMARK"

IfR is noetherian then dim R

supes,e()dimle(P/P)

^{so if R is} noetherian, Proposition impliesthat if every matrix^over R ishomotopic toadirersumof x2matrices then

dim R 1. This may not bethecasewhen Risnot noetherian the next result shows.

Let

K denoteafieldandvavMuationonK with

vMue

group

c

(R,+).

Let

R bethevaluation ring

corrponding

tov. SinceR isanelementarydivisorring

[9],

^{eachmxnmatrixoverR is}equivalent to adiagonM matrix diag{d,...,d} with v(d,) v(+) when d,+ 0 and

d

^{0impli d 0 for}

R

^{j. The}

"elementary

divisors" d d uniquely determine the

equivMence

^cls.

In

this case, following

[4],

^we canexplicitly determine the monoid of homotopy classes.

Let

G⁺

l

and N(a+) {()

= ^n,’n,e

^N,,e

^a+},

^{where N is the set} of nonnegative integers. ^The, N(G+)is anultiplicative monoid with multiplicationinduced from the equation

,

’*,.

For

(.),b(x) N(G+) say () b() if thereexists positive integers rand with r() sb().

It

isesy to check that is a congruenceon

N(a+). Let

PN(a+)

N(a+)/

^{Then PN(G}+) ^{is a} monoid whichcanbe identified with the primitivepolynomialsin withexponents fromG

+. Let

I()be the congruencecls inPN(a+)

represented

by,()

N(a+). To

the

homotopy

^clsin (R)

represented

668 F. DeMEYER AND H. KALHAIL

by the lxl matrix transformation (d)^overRwecanassignthecongruence class

Izv(dl

in PN[s.].

Our

text resultisthatthis assignment extendstoan isomorphism.

PR.OPOSITION

2. IfR isavaluationring corresponding toavaluationvon^afield K witlvalue groupGthen M(R) PN(G

+).

PROOF: Lemma

2and Proposition 3

(l)

^of

[4]

^imply any⁰ Ihl, M(R)^{containsan xrmatrix} transformation diag{d,,...,d} whered, 0and v(d,) v(d,+) for all ^i.

Let

(R) PN(G+) by (Ihl)

=

^x(a’).

^{We only}

^{check is} well defined, then the rest of the

argument

^is routine. If Idiag{d, a}l Id-a{Y, Y,}Iin M(R) with

v(I)

v(I+) for all^jthen

by

Proposition

3(9)

^of

[,1],

diag{d d}@diag{1 ,0 0}isequivalenttodiag{y y,}@diag{1 1,0 0}.

Let

be the entries indiag{d dr}with pairwisedistinct valuations.

By

uniquenessofinvariant fa.ctors i,a,elementarydivisorring, theentries indiag{I I, withdistinct valuationsare

f’, y.

where

,,{f) ,,(d)

^{whe,, weorder}

d,f

^so

v(a’,)< o(d’,+)

^and

^o(f)

<

,,(/+)

^foralli.

^Let ,.,

#{djlv(dj)=

,,(d’,),

S

J

S "} ^and

.,, #{Yl"(/,) v(/;)l S J S

s}. Then @(Ihl)

IE=t ",zta")l

and @(Ihl)

I=

Moreover, by

uniqueness ofinvariant

factors,

pr, qs,

S

5 ^{k so}

I,= ^r,a;)l I,= s,x"l;l

ⁱ

PN[x’]. ’l’hus 4 ^is well defined.

The following^isneededto provea

generalized separated

^{divisorthrem.} Undefined termiology

can betbund in

[7].

LEMMA

3.

Let

RdenoteaPrfifer domain of finite character whose valuation rings at ma.ximal ideals are pairwise independent. If0#^{L is an} ideal in R then thereis afactorization L

=t

^L,

where each L, is contained in

exactly

one maximal idealP, ofRandP,

# P

^if

^#

^).

PROOF:

Since 0#^{L and} R has finitecharacter, L is contained in

only

finitely many naxinml ideals P,,...,PofR.

Let

L, LRp,flR(1G ^k). Theorem4.10 of

[7]

^impliesL

flptaasp,c(n)(LReR).

Since LRp Rp ifL P we have L

,I(LRp,

R)

O,IL, We

always have ^{L, CP,}

Let

v,,v be valuations corresponding to

P,,P respectively.

^{Sincethese valuationsare} pairwise independent, Theorem 22.9

(2)

of

[7]

^{implies that for each 0}

#

^{z L thereis an a} Re,

Rp,

with v,(a) and u(a) ^0. IfS R- P,O then anelementaryexercisegives, S

-

^{R Re,}

^Rp,

^so after clearing the denominator wecan sumea R. Thusa LRp, R L, buta

P

^soeach Li is contained exactlyonemaximal ideal of R.

Morver,

L,

+ L

^Rwhenever

^#

^{jsince L,}

^{+ L#}

^iscontained in no maximal ideals ofR. Thus L

=L, H,

^L,.

^To

^{check uniquens sumeL}

H= ^L

^where

each

L’

^iscontained in exactlyone maximalideal

P

^ofRand

P ^# P’if

^j

^#

^q.

^From

^theabovew

andafter relabeling we can sume

P Pj. Now

LRp,

(= ^{Lj)Re, LjRp,}

^so

^if,

^{Re, L,Rp,. Since}

L

LRp,Ritfollows that

L’,

CL,

But

Rp,

L’,

Rp,L, and

RpL’,

RpL, Rpif P is^amaximalideal of R notequalto so Rp

Li/L’

^{0forall maximalideals PofRso}

L L’

i.

Following [13],

^theideals L, in

Lemma

3 arecalled the

separated

divisorsof L. The

separated

divisors Div{d,}=t ofafinitesequence ofideals in R is the collection, counting multiplicity, of^allthe sepa.rated divisors ofthe individual ideals J,.

For

conveniencewelistsomedefinitions andaresult weneedfrom

[2]. ^A

^Prfiferdomain R with quotient field K is saidtosatisfy theInvariant

Factor Threm

iffor any finitely

generated

^submodule

M of%"} there exist simultaneousdecompositions of

R

^")and M

R

" ’ ... ^{-t Jz} ...

M Elz^{q E,._Iz,._q)E,.z,.}

where

,

K, theJ, areinvertiblefractionalideals of R, the^{F-,i are invertible}integralidealsofRand E,C E,+ for 1,2 1.

A

Preferdomain R has the Steinitz property if for fractional ideals

GENERALIZED EQUIVALENCE OF MATRICES OVER PRUFER DOMAINS 669

and J, qJ

_

^{R 1.I.}

A

Prfifer domain R has the

11/2

generator property in case fi)rany fi’aclioal itleal landany0#zt lthereisayt lwitl l=+Ry.

PROPOSITION

4. If R is a Priifer domain of finite character or Krull dinmnsion tlc gent’ral ll satisfies the Inva.riant

Factor

Theorem, R has the Steinitz property and R h the

15

property

PtOOF: []

SEPARATED DIVISOR THEOREM (Levy). Let

R be^aPrfifer dotninand

A

^anmx mtrix over R ofrankr.

Let Ma

^{be thesubmoduleof R(}

" ^genera

^{by therowsof}

^A

^{and let}

^Sa

I. It’R

is of finite character or direr then there exist invertible ideals

E E

^{of R witl}

E,CE,+(1 1)and invertible fractional ideals

, O+

^{0,, such that}

=

^R/E,

^{+ J+ +... +}

^{d. if ,"<}

^,,

=,

R/E, if ^,.=

,

wlered7 ==E,

^ifr=,n,

=JRifr=Oand

^J,Rif,’=n.

[[. I["/t isof finitecharacterwith pairwiseindependentvaluationringsat maxitnal idealsand

A,A’

are two^,,x,matricesoverR of rank then

A

isequivalent to

A’

if andonlyifDv{E,}’"

Dio{k.’}’=

PROOF: I. Let

M be any finitely

genert

submodule ofRt". Since R satisfies the hva.riat

Factor

Threm thereexistsimultanusdecompositions ofR" and M,

M

EI

^{E__ E,.Jr,.}

witha-, K quotient field ofR, theJ, invertible fractional ideals ofR and E, invertibleintegralitleals of R such that

, c

E,+ < <r-1. Since R satisfiesthe

1 generator

property, Proposition ^of

[2]

implies Jr/ErJr R/Er.

Hence

(where

_d+

...

^{d,, do notoccurif n).}

^Here

^{ris the rankofM whichis therankof}

^A

^if

^m

^is

generated bytherowsof

A.

This prov

SA

h the decomposition giveninI. If then

so

E ..

^E,__

^EJ

^{"’). TheSteinitz}

^Property

and ccellation imply

=

^{E,.I,. R}

and

Jy =l

^E,.

^In

^{thesame way, ifr 0then R") d}

...

^{J,, andJ...J, J,+...J, R.}

If then R^’)

R ...

^{R_ J,so}

^J

^R.

I. Let

A,

A’

be two x nmatricesoverR.

Suppose

wehavethedecompositions given in for

Sa

^and Sa, If

A

is

uivalent

^to

A’

^then

Sa Sa,

The uniqueness

part

oftheInvnriant

Factor

Threm

(s [12])

i,npli Div{E,}=,

Div{E},

and J,+...J,

J+ ^...J’,. Conversely,

if Div{E,}=,

Oi,,{E}’,’=,

and

J+

^...J,

d+ ...J

^then

Sa

^and Sa, havethesameinvariant factors and thus areisomorphic.

To

completethe proofof

II

weneed to checkthatif

Sa Sa,

^then

A

^isequivalentto

A’. Our

problem is to findisomorphisms

,

^0,

a

making the commuting

diagram

Rm

^Rm)

_Sa

0

o A’ Snce

R 8

re,

^beorem 1.8

o [9

mphe8

.

^{? 6ere?8}

^poectve

^and

60 F. DeMEYER AND H. KAKAKHAIL

If SA ^{P is}the projection let pbeasplittingmap^{so R("} p(P)+Qand ker ,1CQ. It thesame way Sa,

P’+T’,

R⁽⁾

p’(P’)+Q’

and ker

1’

^C

^Q’.

^Since

Sa Sa,

^thereare

isomorl)hists

^a: !’--

!"

aud 7’+

7". By

Proposition of

[12],

^{T isadirectsumof cyclicmodules} R/L^{for ideals0 L}C 1.

By

lean,ha3 wecan let {L,} bethe setof separateddivisorsofL. The ChineseRetnain(h’r

implies R/L +R/L,. SinceL, is contained in

only

onemaximalideal of R, R/L, is local tbr all ^i.

Since Q/kero 7",Theorem 1.6of

[11]

^{impliesthere isasimultanusdepositiouof Qan(I ker 0}

In

thesamewaythereisasimultanusdecompositionof

Q’

andker

0’.

Thus the given isomorpltis

T

7"

extends to an isomorphism :Q

Q’

such that _+(ker0) ^ker

0’.

^Thisgives isom<>rpliss

+

^and

+ ^p’a +

^makingthe commutativediagram

R^(M) R^(M)

SA

0

Sillc(’07(ker 0) ker,l’ wehavetheexact

R(

’A’

IrnageA’---.O

ald

:,

is the lt-honomorphism given since R

’’

^{is free. Thiscompletesthe proofof the}

^Separated

Divisor Theorem.

3.

THE MONOID OF HOMOTOPY CLASSES

LEMMA

5.

Let

RbeaPr/iferdomainoffinitecharacterorKrulldimension <1.

(1)

If0

# Ill

^{.(R) then thereexists g such that}

Ill

^{Igl in} sM(R) and coker(g) ^{is a torsio} R-module.

(2) Let

Istl, lgle ./I(R} with

coker(g)

and

coker(st)

^{torsion R-modules. If M g g P Q}

theu I/I Igl in .X4(R) if andonly ifthereexists

R-progenerators

K,L such that ^g(R)If Q63L and

coker(f)

(R)K coke,’(g)(R)L.

PROOF:

The proofnowfollows "mutatis mundantis" asthe proofof lemma 6of

[4].

Following page393 of

[4]

^adescriptionof Yt4(R) in terms of ideals of Rcanbe givennow. Consider the set

A

{(M,R(’))Im

>1,M isafinitely

generated

Rsubmodule of R^(m)such that

R("*)/M

is torsion Rmodule.

Definemultiplicationin

A

asfollows: (M,

R(m))(N,

R

(n))

(M^(R)N,R(m) (R)

R(n)),

then

A

isacommutativc semigroup. Defineasemigroup homomorphism

p:A--..’(R) A4(R)-{0}

byp(M,R

(’)) Jil

^{where M R(’) is} the inclusion map.

Note

that by

Lemma

5 p(N,

R("))

if and

only

ifthereexist

R-progenerators

Kand Lsuch that

R(’) (R)K

R(’)(R)

L and

(R(m)IM)(R)

K

- (R(")IN)(R)

^L.

p(M,R("))

Thust)inducesanequivalence relation on Aasfollows:

(M,R(m))~(N,R(n))

ifand onlyif

p(M,R(m))=

p(N,R

(n))

GENERALIZED EQUIVALENCE OF MATRICES OVER PRUFER DOMAINS ⁶⁷¹

Let

.4bethesetofequivalenceclasses of

A.

Then the

product

on

A

induces themultiplication^o,.4, turningitintoacommutative monoid with identitythe class containing(R,R).

LEMMA

6.

Let

R beaPrfifer domain of finitecharacteror Krulldimension < 1, then tlw nlap

"

^.4

.

(R) induced by pis anisomorphism.

PROOF: Same

as Proposition 7 of

[4].

THEOREM

7. IfR is a Priifer domain offinite character or Krull dimension < then every homomorphism of

R-progenerators

ishomotopictoamatrix transformation of theform

0 a: b:

am bm ^{m 2m} where

if/

=(a,b) then

I

3l+(l<j<n-1).

PROOF:

The proof of Theorem7 now followsexactly ason pages 391-394of

[4].

LEMMA

8. Let Rdenotean integraldomain. Then thefollowingareequivalent.

Eachnonzeroelementof R is contained inonly finitelymany maximal ideals ofR.

2)

^{IfN is a finitely}

^generated

submodule of^R(") then

Re

^{(R)N is a direct}

^summand

^of

R

^{") for}

almost all PeMaxSpec(R).

(3) For

^eachexact sequence 0 ^N Q

m

0of finitely

generated

R-moduleswith Qprojective’

the associated sequence 0 Rt,(R)N R,(R)Q Rp(R)M 0is

split

exactfor almost allPeMa:cSpec( /i’)

PROOF:

The equivalenceof2and3 followseasilysince everyfinitely

generated

projective naodule isadirectsummand ofafreemodule offiniterank. Tosee 2 let0 a R. Then

(a)

^isasubmodule

ofRso (R) iseither0or aunit in

Re

^{foralmost all} PMaxSpec(R). IfRisan integraldomain then

(R)a 0 in

Re

^{so a P for almost all} PeMa:Spee(R).

To

show 2 let R(")

Rx ...

^{$R,,, and}

let N

Rn

+...

+

^{Rn, with0 n,}

R(n}(1 _< _<

k).

Let

n a+...

+

^{a,:, where}we can assume a ^0.

Over a’(R

^wehave

a[n,x,.

xn^{is a}basisfor

(a-(R)

^{") and}

a-{n,n:,..,n: generates a’N

so

replace x

^and

n

by

a{n

^over

^ajaR.

^{Then a,,...,,, is a basis for}

(a’R)

^{n and z,n.}

generates

a?

^{N. Writen. bz}

+

b.z

+ +

^{b,x,. Since}ze

a[N

^wecan

replace

n by b.z+...

+

where be ^{0 or ,e}

a’[Rn a’{R:.

If b. ⁰then over

ba-(R

^wehave

,bn.,:z ,,

isabasis for

(b.a-[R)

^") and

z,btn2,nz

^nt, generates

baN

^so

^replace

^{and n by}

bn

^over

b.a’R.

Then N

(ba-{

R)x

+

(b. a’{ R)x

+

(b a-{ R)nz +...

+

(b a-{ R)n.

Afler

^finitelymanystepswe can find anelementc R withc-N adirect summandof/i5"). Sincecis inonly finitelymany naaxima.l idealsofRby 1, Re^{(R)N isadirect}summand of

R

^{’*)for almost all}PMa:Spec(R).

IfR is an

integral

domainof finite character then a nonzero homomorphism

m

N of R- progeneratorsinducesan

image

split

homomorphism R,(R)MX-,IRt,(R)N

for almost all

PeMaxSpec(R)by

3 of

Lemma

8.

In

thiscasetheinclusions R

Re

induceanaturalmap :M(R)

LEMMA

9. If R isa Prfifer domain of finitecharacter

and

M(R)

,,M,,s,n)M(R,)

^is inducedfrom the inclusions R Rp then isamonomorphism.

PROOF: Let

0

Ill

^{0 Igl be}in M(R)andassume(I/’1) (Igl). Then [1^(R)

II l1

^(R)

for all PMaxSpec(R).

By

Theorem 7, ⁰

Is’l

^{determines adescendingsequence 1}

I.

^{::)1,, of} ideals in R with 0

I

^{(a,b) and 0} Igl determines

J or,,,

with 0

or,

(c,,d,). Under

j R,,)

the isomorphism_#of

Lemma

6,

(’]=I

^R")corresponds to

Ill

and (9=

,

correspondstoI.ql-

We

show

(’]=I,R

^’)

(or,R)

ⁱⁿ A

Let

P be a maximal ideal for which

II

^(R)

672 F. DeMEYER AND H. KAKAKHAIL

is split.

In

A(Rp) the split class is represented by (Re,Re) and projective modules ^{are free over}

Re

^so

(R(’)/=t

I)^tt)(R)

Re

(0) which implies

I

^{c Pfor j 5 n.}

^In

^{thesame way}

^{y c}

^{P so}

lRp ^Rp

JRe

^{for all i,j whenever}

11 fJ Jl l

^{issplit over}

Re.

^{Consider thefinite set}

(si,ce

R h finite

character)

^S {PMazSpec(R)II1

fl

Ila.I}

{PMazSc(R)II1

@

l #

Iln.I}. Since

17= ^e,,ng’l IZ neJ,nl

^{in A(Rp) for eh P S there are} positive integers se,te with

spn tpmand

[Rp/Rpb]t’"

[Rp/RpJ]^).

Let

^{He,ssp and p,stp.} Then for each P S,

l[Rp/Rpl2]{,) [,=lRe/RpI,](,)n _ [t=lRP/Rejt](,)m

,=1

[RP/Jk]

^{(t)" The ideals Rt,}

I

^and

ReJk in thisdecompositionare

uniquely

determinas sets

(ii

pg 260 of

[12]).

^{This means thelist}

ofideals RpIt

Re I

^and

^{RJ RFJ...}

^where

It I

^and

^{J J}

^{are thesa,c for}

each ^p S. Therefore,foreachj,t,

P,Ms,,n)(J Re

R) ^forcorresponding k,^w

We have shown

[t(R/I)] ’ ^[=(R/J)]

^so

^(R),=6) (R",=J)

ⁱⁿ A and ^{is a} ,onomorphism.

We

need two ey lemm about valuations^on Preferdomains.

IEMMA

10.

Let

R be aPrefer domain of finitecharacter whose valuations at ma.ximal ideals re pairwise independent.

Let

p p, be a finite set of maximal idealsofR, let G, be the value group ofRe, and let0<a,G,(1 n). Then there exists^afinitely

generated

ideal of R such that IRe, {o Re, lve,() 9i} and the

only

maximalidealsofRcontaining are

PROOF:

Since the valuations atthe maximalidealsofRarepirwise independent, Theorem22.9 of 7 impliesthere exists anx Re,

... ^Re.

^{such that ve,(x) ,(1 n).}

^As

^weobserved in the

proof

of

Lemma

3, we canchoosez R. SinRhfinitecharterwe can let

O

^{Q,, beall the}

maximal ideals of R distinctfrom {Pi},% such that ^r

O(1 S

^j m). Again, Theorem 22.9 of

[7]

implies thereare u, R withve,(,)

,

and

v,(,)

^{0forallj i,k.}

Let

I (x, u). Then ^is finitely

generated, IRe,

zRe, { Re, lve,(a)

E

9,} and the onlymaximal ideals of R containing re P,

., Pn.

LEMMA

11. Let v,v be valuationson field K with value groupsGt,G2 and valuation rings V,

V

respectively. Ifforchpair(9,)

G

^x

G

with0S and0S⁹

there

^isan

andv(r)=9then thevaluationrings

v, v

^areindependent.

PROOF: (S

9, pg 289 of

[7]).

THEOREM

12.

Let

RbeaPrfferdomain. Theinclusionmaps R

Re

^{induce n}

isomorphism

’(R)

e,,s,,n)(Re)

^ifand

^only

^{if R is} of finite character and the valuation rings at the maximalideals ofRarepairwiseindependent.

PROOF: Assume

RisaPrffer domain of finitecharacter.

Lemma

9giv isamonomorphism.

We

check that if in ddition the valuationrings^atthe maximal ideals of Rarepairwiseindependent then is anepimorphism.

Let

(lgel)e,,s,,<n) be^{anelement of}

e,,s,,n)(Re).

^{Then ae isan} inaage splitmap for all butfinitelymany maximal ideMs

p

pof R. Each

I.1

^canbe

^reprented

^by

adiagonalmatrix

(Proposition 2). By

tensoring thesematric withidentitymatrices ofappropriate sizeswecan sumeeach 19,1is

reprented by

a

diagonM

^mxnmatrix.

Let

19e, be

represented by

diag(a, a,m)(1 k).

Let

v, beavaluationdeterminedby thevaluationring

Re,

^withvalue group G, and let _9, vi(a,)(1 k, j m).

Lena

10 gives finitely

generated

ideals

I

^contained

GENERALIZED EQUIVALENCE OF MATRICES OVER PRUFER DOMAINS 673

in exactly the maximal ideals

P P

^and

IRe,

{a RP,Iv,(a)> g,} for <j<m. I,et

I11

sucl that l/I corresponds tothe element

(

^l,

R(’))

^ofA u {0} ^{under the}isomorphisn 6. Then (Ifl) (lgPl)e,MaSec(R)^SOb is^anepimorphism.

Conversely, ^assume the inclusion maps R- Rp for PMaxSpec(R) induce the

isomorl)lis Let

0 R and let

e,

R R by the honomorphism given by left multiplication by a. ’l’lc

levi lnl

in M(R) if and only ifa is aunit in

n (Proposition 3(5)

^of

[4]).

The inmgc

o1"4

^{will lie}

in t,,MaSpee(a).A(Rp)onlyif the image of

levi

^{in M(Rp) is}

Ilm.I

^foralmost all PM,zSpcc(R). This means a

it

^{Pforalmostall}PMaxSpec(R)soR must havefinitecharacter.

Let

P,Qbetnaximal ideals of R and let S Rpf

RQ.

^{Since isan}epimorphism, theinduced map ,"A/I(S) AI(Rt,)

isanepimorphism.

To

seethe valuation ringsR, and

P

^areindependentwecheck the condition of Iemna 11.

Let

vp and ₀ be thevaluationscorrespondingto valuation rings ^Rpand

groul)s(;p,Gq. ^{I,(’t 0}<.q Gt,and0<_h

GO

^{and let Rp, RQwith vp(a) g,vQtb) h.}

oll(), liere isa

Ihl

^{3A(S)withII(R)hl}

levi

ⁱⁿ

M(Re)and

Ill,

hi Itl

ⁱⁿM(RQ). Since^{Sisas(’ilo(’al} Bczou! donai,

s

isa elementarydivisor domain

[9]

^{so wecan representh bythediagonal nalrix}

(liag(c, ,c,.).

As

we sawin theproof of Proposition 2,we canfind units

ue ^Re

^and

w

^{RQSll(’htlat}

cju in

Re

and

c

^u,bin RQ(I <_^j<_k). Thus vt,(c) vt,(ua) v,(a)and ve(c) VQ(Wlb) vQ(b) which shows theva.luation rings Re and /Q arepairwise independent.

REFERENCES

I. H.

Bass,

K-Theory and Stable

Algebra,

Publ.

I.H.E.S. 22(1964),

pp. 5-60.

"2..I. Brewerand L. Klingler, Pole Assignabilityand theInvariant FactorTheoren for Priit’cr l)o- na.insand Dedekind Domains,

J. ^Algebra.

^Vol.

1.112 ^#2.

^{1987,pp. 536-545.}

3. WillyBrandal,TheCommutative

Rings

^Whose

Finitely

Generated ModulesI)ecoml)OSe,

Notes

in

Mathenaics, Springer-v’erlag,

Vol.

723(1979).

1.

F. R. DeMeyer

and

T. J. Ford,

Homomorphisms of

Progenerator Modules, J.

of Algebra,

!1:,

Number 2, March 1988pp. 379-398.

5.

F. R. DeMeyer

and

E. Ingraham,Separable

Algebras

Over

Commutative

R.ings.,

^Lecture

in Mathematics,

No.

181,Springer-Verlag, Heidelberg, 1971.

6. ’l’.3. Ford, Homomorphismsofprogenerator modulesundera

change

of base ring, (_’,omnmi(’a- tions

i.n ^Al;ebra 1_6(3)(1988)457-482.

7.

R.

Gihner.

Multil)licative

IdealTheory,

M.

Dekker,

New York,

1972.

8.

W.

Heinzer, Quotient

Overrings

of

Integral

Domains, Mathematikh 17

(1970)p.

139-148.

9.

I. Kaplansky,

Modulesover DedekindRings and Valuation Rings,

Trans. Amer.

Math.

Soc.

72

(1952),

327-340.

10. M. D.

Larsen, W. J.

Lewisand T.

S. Shores,

Elementary ^Divisor Ringsand Finitely Presented

Modules, T.rans. ^Amer.

^Math.

^Soc.

^Volume

^{187, Issue}

^{1, 1974 pp. 231-248.}

11.

L.

l,evy, DecomposingPairsof

Modules, Trans. Amer.

^Math.

Soc.

122

(1966),

^{pp. 64-80.}

12. ----, lnvariantfactor Theorem for Priifer Domains of Finite

Character, J.

of

Algebra,

Vol. 106, 1987,pp. 259-264.

13. ----,Almost

Diagonal

Matrices

Over

Dedekind Domains, Math

Z.

124

(1972),

pp. 89-99.

Frank

DeMeyer

Department

ofMathematics Colorado

State

University Fort Collins,

CO U S A

80523

Hainya Kakakhail 21A VictoriaPark The

Mall,

Lahore Pakistan

Mathematical Problems in Engineering

Special Issue on

Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios

Call for Papers

Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.

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