A Principal Ideal but not Euclidean Domain

7/24/2019 A Principal Ideal but not Euclidean Domain

1/6

A Principal Ideal Ring That Is Not a Euclidean RingAuthor(s): Jack C. WilsonSource: Mathematics Magazine, Vol. 46, No. 1 (Jan., 1973), pp. 34-38Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2688577.

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2/6

34

MATHEMATICS

MAGAZINE

[Jan.-Feb.

terminal

ositions.

he first election

f a turning

rc must

be at the largest cute

angle

n

order

ominimize he arc,and hence,

o minimizehe

path.

B

a'~~~~B

FIG. 7. Crossed

oins,

one

pivot oint.

6.

The

restricted

roblem

with rossed

oins

and close terminal

ositions.

n each

oftheforegoingases,the motion s composedof a combination frotations nd

translations.

igure 8, however,

hows

a case in which

AA'

crosses

BB',

and

the

positions

AB

and

A'B' are so close

that

tangents

annot

be

drawn

to

the

turning

arcs

as

in

Figures

6 and

7.

Then,

t is

necessary

o use

pure

rotation

f

the

ine AB

about

a center

which s

at the

intersection

f the

perpendicular

isectors

f

AA'

and

BB'.

B AA

FIG.8.

Crossed oins,pure

otation.

A PRINCIPAL

IDEAL

RING

T HAT IS NOT

A EUCLIDEAN

RING

JACK

C.

WILSON,

University

f North

arolina t

Asheville.

1. Introduction.

n introductorylgebra

texts

t

is

commonly roved

thatevery

Euclidean

ring s a principal

deal ring.

t is

also

usually

tated

hat

the

converse

s

false,

nd the student

s

often

eferredo

a

paper

by

T. Motzkin 1].

Unfortunately,

thisreference oes not containall of thedetailsof thecounterexample,

nd it

is

not easy

to find

heremaining

etails

from he

references

iven

n

Motzkin's

paper.

The object

of

this article

s to present

he counterexample

n complete

detail

and

in a form

hat

s accessible

o students

n an

undergraduate

lgebra

class.

Not

all authors

use precisely

he

samedefinitions

orthese

two

types

of rings.

Throughout

his

paper

the following

efinitions

ill

hold.

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3/6

1973] A PRINCIPAL IDEAL RING THAT IS NOT A EUCLIDEAN RING 35

DEFINITION 1. An integraldomain R is said to be a Euclidean ring ffor very

x

:

0

in R

there s

defined nonnegativenteger (x) such that:

(i) For

all x and

y in R,

both

nonzero, (x)

n}

.

This

defines he

sequence

which obviously as propertiesi)

and

(ii). Suppose

thatx is in

Pn

and

y

is in

Ro.

d(xy)

?

d(x)

>

n which

mplies

hatxy is in

P,.

This

shows that

P, Ro

P,,;

i.e.,

foreach n, P,, is a product deal.

For

propertyiii)

let x be in

Pn; i.e.,

x is in

P,

and there xists

y

in R such that

y +

xR

P,n.

Applying

he Euclidean

algorithm,

here xist lements and r in

R

with

y =xq

+

r and r

=

0 or

d(r)

d(r) _ n1,

so

that

d(x)

>

n

+

1

and

x

is

in

P,1

.

This proves property iii)

P

Pn+1

For

propertyiv), clearly

Ro

=

P0

and

application

of

(ii) gives Ro

= P'

c

P1.

Assuming

hat

R() cP,,

Lemma

2

and

(iii) yield

R(

+1)c

P

c

P,,

.

By

induc-

tion, iv)

is

proved.

COROLLARY. If Ro

=

R'o

0,

then

R is not a Euclidean

ring.

Proof.

The hypotheses f the corollary

mply hat for all n,

R(") = R'.

If

R

is

a

Euclidean

ring,

the

theoremwould

require Ro

=

n R(")

cf

n

P,1

0.

This corollary s

now

used to show that R is not a Euclidean ring. First

Ro is determined.f x is a unit in R, say xy = 1, and z is an elementof

R,

z +

x(-yz)

=

0 is not

n

Ro.

Thisshowsthatunits re not nRo. If x is nota unit

in

R,

then

using

z

=

-1,

z

+ xy

:

0 for

all

y

in

R,

which hows that

f x

is

not

zero and nota unit,x is inRo. Altogether, ' is precisely he set of elements f R

that

are neither nits

nor

zero.

Notice that the

only units

of

our

example R are

1

and

-

1.

Next,

n

order

to

determine he elements f

R",

it s convenient

o

use the

following erminology:

DEFINITION

5.

An

elemenit of R' is said to be a side divisor

J

y in R

pr-ovidedl

ther-e

s

a

z

in

R

that s

not n Ro such that x| (y + z). Anz lement

x'ofR0

is

a

universal

side divisor

provided

hat

it is

a

side

dlivisor

f

everyelement

f

R.

If x is in R", then x is in R' and there s a in R such that v +

xC

R

'

i.e.,

x

never

divides

y

+

z

if z is

zero or

a

unit.Thus, x

is not

a

side

divisorof

y,

and

therefore,

ot a universal

ide divisor.Conversely,

f

x

is

not

n

R",

and

is

in

Ro,

then for

every y

in

R thereexists a

w

in

R

with

y +

xw

not in

R'; i.e.,

y +

xw

is zero or

a

unit,

nd

therefore,

is

a side

divisor

f

y. Since

thisholds for

everyy

in

R,

x

is a

universal ide

divisor.Together,

hese two

arguments

how

that R'"

is the set R'

exclusive

of the

universal

ide

divisors. f

it

can now be

shown

that R

has no

universal ide

divisors, his will show that R'

=

R"o

#,

,

and the

corollary

will

complete

he

proof

thatR is not a

Euclidean ring.

A sidedivisorof 2 in R mustbe a nonunit ivisorof 2 or 3. In R, 2 and 3 are

irreducible,

nd

therefore,

he

only

side divisorsof 2 are

2, -2, 3,

and

-3.

On

the other

hand,

a

side

divisor

of

(1

+

/-19)/2

mustbe a nonunitdivisorof

(1

+

1-19)/2, (3

+

/1-19)/2,

or

(-1

+

1-19)/2.

These elements of

R

have

norms

f 5, 7, and 5, respectively,

hilethe normsof

2

and 3 and their ssociates

are

4 and 9, respectively. s

a

result,

o

side divisor

of

2

is

also

a

side divisor

of



6/6

38

MATHEMATICS MAGAZINE

(1 +

1/-19)/2,

and there

re no universal ide

divisors

n

R.

All of the

details of

the counterexample

re complete.

References

1. T. Motzkin, he Euclidean lgorithm,

ull.Amer.Math.

oc., 55 1949)1142-1146.

2.

H. Pollard,

he

Theory

f

Algebraic umbers,

arus

Monograph , MAA,Wiley,

ew

York,

1950.

THE U.S.A. MATHEMATICAI, OLYMPIAD

Sponsoredby

the

MathematicalAssociation f

America,

he first

.S.A.

Mathe-

matical Olympiad was held on May 9, 1972. One hundred tudents articipated.

The eight op rankingtudentswere:

James

axe, Albany,

N.Y.;

Thomas

Hemphill,

Sepulveda, Calif.;

David Vanderbilt, arden City,N.Y.;

Paul

Harrington, entral

Square, N.Y.; Arthur ubin,

West

Lafayette,nd.; David

Anick,New Shrewsbury,

N.J.; StevenRaher,

Sioux City, owa; James hearer, ivermore, alif. A detailed

reporton the Olympiad ncluding he problems nd

solutionswill appear in the

March 1973 issue of

the American MathematicalMonthly.

The second U.S.A. MathematicalOlympiad will be

administered n Tuesday,

May 1, 1973. Participation s by invitation nly. For

further articulars, lease

contact the Chairman of the U.S.A. MathematicalOlympiad Committee,Dr.

Samuel

L. Greitzer,

MathematicsDepartment,Room

212,

Smith

Hall,

Rutgers

University,Newark,

N.J.

07102.

GLOSSARY

KATHARINE

O'BRIEN,

Portland,Maine

Campusdisorder

Skew

quadrilateral-quadrangle

extreme

olarization

demonstration.

Generation ap

Two-parameter

amily

negative rientation

to communication.

Heart

transplant

Removable

discontiniuity

binary orrelation

operation.

Nylontires

Synthetic

ubstitution

translation

y

rotation

transportation.

Popcorn

Iterated

kernels

onto

magnification

transformation.

Suburb

Deleted

neighborhood

littlenclination

to

integration.


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