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
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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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