A Principal Ideal Ring That Is Not a Euclidean Ring

A Principal Ideal Ring That Is Not a Euclidean Ring
Author(s): Jack C. Wilson
Reviewed work(s):
Source: Mathematics Magazine, Vol. 46, No. 1 (Jan., 1973), pp. 34-38
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2688577 .
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34
MATHEMATICS MAGAZINE
[Jan.-Feb.
terminalpositions.The firstselectionof a turningarc mustbe at the largestacute
anglein orderto minimizethearc, and hence,to minimizethepath.
B
a'~~~~B
FIG.
7. Crossedjoins,one pivotpoint.
problemwithcrossedjoins and close terminalpositions.In each
6. The restricted
of the foregoingcases, the motionis composed of a combinationof rotationsand
translations.Figure 8, however,shows a case in whichAA' crosses BB', and the
positionsAB and A'B' are so close thattangentscannot be drawnto the turning
arcs as in Figures6 and 7. Then, it is necessaryto use pure rotationof the line AB
of the perpendicularbisectorsof AA'
about a centerP whichis at the intersection
and BB'.
B
AA
FIG. 8. Crossedjoins,purerotation.
A PRINCIPAL IDEAL RING T HAT IS NOT A EUCLIDEAN RING
of NorthCarolinaat Asheville.
JACKC. WILSON, University
algebratextsit is commonlyprovedthatevery
In introductory
1. Introduction.
Euclidean ringis a principalideal ring.It is also usuallystatedthatthe converseis
to a paper by T. Motzkin[1]. Unfortunately,
false,and thestudentis oftenreferred
and it is
this referencedoes not contain all of the details of the counterexample,
givenin Motzkin'spaper.
not easy to findthe remainingdetailsfromthe references
The object of this articleis to presentthe counterexamplein completedetail and
in a formthatis accessibleto studentsin an undergraduatealgebraclass.
for thesetwo typesof rings.
Not all authorsuse preciselythe same definitions
hold.
will
definitions
Throughoutthispaper the following
1973]
A PRINCIPAL IDEAL RING THAT IS NOT A EUCLIDEAN RING
35
DEFINITION 1. An integraldomain R is said to be a Euclidean ring ifforevery
x : 0 in R thereis defineda nonnegativeintegerd(x) such that:
(i) For all x and y in R, bothnonzero,d(x) < d(xy).
(ii) For any x and y in R, both nonzero,thereexist z and w in R such that
=
x zy + w whereeitherw = 0 or d(w) < d(y).
2. An integral domain R with unit elementis a principal ideal
ring if everyideal in R is a principal ideal; i.e., if everyideal A is of theform
A = (x) for somex in R.
DEFINITION
The ring,R, to be consideredis a subsetof thecomplexnumberswiththeusual
operationsof addition and multiplication:
R = {a + b(t + 1-9)/21 a and b are integers}.
to showthatR is an integraldomainwithunitelement.The purpose
It is elementary
of thisarticlethenis to showthatR is a principalideal ring,but thatit is impossible
to definea Euclideannormon R so thatwithrespectto thatnormR is a Euclidean
ring.
2. The ringis a principal
idealring.In R thereis theusual norm,N(a + bi) = a2 +
b2, which has the propertythat N(xy) = N(x) N (y) for all complex numbers
x and y. In R thisnormis alwaysa nonnegativeinteger.The essentialtheoremfor
this part of the exampleis due to Dedekind and Hasse, and the proof is taken
from[2, p. 100].
THEOREM 1. Iffor all pairs of nonzer-o
elementsx and y in R withN(x) ? N(y),
eitheryI x or thereexist z and w in R with0 < N(xz - yw) < N(y), thenR is a
principal ideal ring.
Proof. Let A : (0) be an ideal in R. Let y be an elementof A withminimal
nonzeronorm,and let x be any otherelementof A. For all z and w in R, xz - yw
is in A so that eitherxz - yw = 0 or N(xz - yw) > N(y). Hence the assumed
conditionson R requirethatyI x; i.e., A = (y).
The ring R under considerationwill now be shown to satisfythe hypotheses of Theorem t. Observe that 0 < N(xz - yw) < N(y) if and only if
0 < N[(x/y)z - w] < 1. Given x and y in R, both nonzero and ytx, writex/y
in the form (a + bV-19)/c where a, b,c are integers,(a, b,c) = 1, and c > 1.
First of all, assume that c > 5. Choose integersd,e,f,q, r such that ae + bd +
cf = 1, ad- 19 be = cq + r,and I rl ? c/2. Set z=d + ej-19 and w=q-fJ -19.
Thus,
(x/y)z-w = (a + bJ-19)(d + e 19)/c--(q-f
/-l9)
= r/c+
-19/c.
This complexnumberis not zero and has norm(r2 + 19)/c2,whichis less than I
obvious is c = 5,
sinceIr ? c/2and c ? 5. The onlycase thatis not immediately
<
r2
2 so that + t9 < 23 c2.
but then Ir
_
36
MATHEMATICS MAGAZINE
[Jan.-Feb.
The remainingpossibilitiesare c = 2, 3, or 4. Consider these in order:
(i) If c = 2, y, x and (a, b,c) = 1 implythat a and b are of oppositeparity.
Set z = 1 and w = [(a -1) + bVI-19]/2 which are elements of R. Thus,
(x/y)z - w = 1/2 # 0 and has norm less than 1.
(ii) If c = 3, (a, b,c) = 1 impliesthat a2 + 19b2 _ a2 + b2 0 0 (mod 3). Let
and w = q where a2 + 19b2 = 3q + r with r = 1 or 2. Thus,
z = a - b-19
(x/y)z - w = r/3: 0 and has norm less than 1.
(iii) If c = 4, a and b are not both even. If they are of opposite paO (mod 4). Let z = a-b /-19 and w =q, where
rity,a2 + 19b2 _ a2 -b2
a2 + 19b2 = 4q + r with 0 < r < 4. Thus, (x/y)z-w = r/4:# 0 and has norm
less than 1. If a and b are both odd, a2 + 19b2 -a2 + 3b2 X 0 (mod 8). Let
z = (a - bV-19)/2 and w = q, wherea2 + 19b2 = 8q + r with0 < r < 8. Thus,
(x/y)z - w = r/8: 0 and has norm less than 1.
This completesthe proof that R is a principalideal ring.
is takenfrom
3. The ringis nota Euclideanring.Thispartof thecounterexample
[1]. The materialis repeatedand slightlyelaboratedhere in order to give a selfcontainedresultaccessibleto an undergraduateclass. As withthe previoussection
the resultsare statedwithinthe contextof the ringR underconsideration,but the
theoremapplies to more general integraldomains. Throughoutthis section Ro
will denote the set of nonzero elementsof R.
3. A subsetP ofRo withthepropertyPRO(P; i.e., xy is an element
of P for all x in P and y in Ro, is called a productideal of R. (Notice that Ro
is a product ideal.)
DEFINITION
denotedby S', is defined
DEFINITION 4. If S is a subsetofR, thederivedset ofJS,
by S' = {xeSj y + xR c S, for some y in R}.
LEMMA
1. IfS is a productideal, thenS' is a productideal.
Proof. If x is in S', thenx is in S and thereexistsy in R such thaty + xR c S.
Let z be in Ro. Since S is a productideal and x is in S, xz is in S. Further,
y + (xz)R e y + xR e S. This shows that S'Ro e S'; i.e., S' is a product ideal.
LEMMA
2. If S c T, then S' c T'.
Proof.If x is in S', thenx is in S and hencein T, and thereexistsa y in R such
thaty + xR c S c T. Therefore,x is in T', and S' c T'.
THEOREM 2. If R is a Euclidean ring, then thereexists a sequence, {P,}, of
productideals with thefollowingproperties:
D ,
(i) Ro = PO DP1 DP2 D' D P
=
(ii)
0,
P,
for each n7, and
(iii) PI c
the nth
(iv) For each n , dnerived
set of Ro, is a subsetof P,,
,
Proof.Let theEuclidean normin R be symbolizedby d(x) forx in Ro. For each
1973]
A PRINCIPAL IDEAL RING THAT IS NOT A EUCLIDEAN RING
37
R d(x) > n} . This definesthe sequence
nonnegativeintegern, defineP, = {x eRo
which obviouslyhas properties
(i) and (ii). Suppose thatx is in Pn and y is in Ro.
d(xy) ? d(x) > n whichimpliesthatxy is in P,. This shows thatP, Ro ' P,,; i.e.,
for each n, P,, is a productideal.
For property
(iii) letx be in Pn; i.e., x is in P, and thereexistsa y in R such that
thereexistelementsq and r in R
y + xR P,n.ApplyingtheEuclidean algorithm,
with y =xq + r and r = 0 or d(r) < d(x). Hence, r = y + x(-q) is in
v + xR c P, whichimpliesthat d(r) _ n, and in turn,d(x) > d(r) _ n1, so that
d(x) > n + 1 and x is in P,1 . This proves property(iii) P Pn+1 .
For property(iv), clearlyRo = P0 and applicationof (ii) gives Ro = P' c P1.
Assumingthat R() cP,, Lemma 2 and (iii) yield R( +1)c P c P,,+. By induction, (iv) is proved.
COROLLARY.
If Ro
=
R'o : 0, thenR is not a Euclidean ring.
Proof. The hypothesesof the corollaryimplythat for all n, R(") = R'. If
R is a Euclidean ring,the theoremwould require Ro = n R(") cf n P,1= 0.
This corollaryis now used to show that R is not a Euclidean ring. First
Ro is determined.If x is a unit in R, say xy = 1, and z is an elementof R,
z + x(-yz) = 0 is not in Ro. This showsthatunitsare not in Ro. If x is not a unit
in R, thenusingz = -1, z + xy : 0 forall y in R, whichshows thatif x is not
R' is preciselythe set of elementsof R
zero and not a unit,x is in Ro. Altogether,
that are neitherunitsnor zero. Notice that the only unitsof our example R are 1
and - 1. Next, in orderto determinethe elementsof R", itis convenientto use the
followingterminology:
DEFINITION 5. An elemenit
x of R' is said to be a side divisoroJy in R pr-ovidedl
ther-eis a z in R that is not in Ro such that x| (y + z). Anzelementx'ofR0 is a
universal side divisorprovidedthat it is a side dlivisorof everyelementof R.
If x is in R", then x is in R' and thereis a ! in R such that v + xC R '
i.e., x neverdivides y + z if z is zero or a unit.Thus, x is not a side divisorof y,
and therefore,
not a universalside divisor.Conversely,if x is not in R", and is in
Ro, then for everyy in R thereexists a w in R with y + xw not in R'; i.e.,
x is a side divisorof y. Since thisholds for
y + xw is zero or a unit,and therefore,
everyy in R, x is a universalside divisor.Together,these two argumentsshow
that R'" is the set R' exclusiveof the universalside divisors.If it can now be
shown that R has no universalside divisors,this will show that R' = R"o #,
0,
and the corollarywill completethe proofthatR is not a Euclidean ring.
A side divisorof 2 in R mustbe a nonunitdivisorof 2 or 3. In R, 2 and 3 are
the only side divisorsof 2 are 2, -2, 3, and -3. On
irreducible,and therefore,
the other hand, a side divisor of (1 + /-19)/2 must be a nonunitdivisor of
(1 + 1-19)/2, (3 + /1-19)/2, or (-1 + 1-19)/2. These elements of R have
normsof 5, 7, and 5, respectively,
whilethe normsof 2 and 3 and theirassociates
are 4 and 9, respectively.
As a result,no side divisorof 2 is also a side divisorof
MATHEMATICS
38
MAGAZINE
(1 + 1/-19)/2, and thereare no universalside divisorsin R. All of the details of
the counterexampleare complete.
References
Bull.Amer.Math.Soc., 55 (1949)1142-1146.
1. T. Motzkin,The Euclideanalgorithm,
CarusMonograph9, MAA,Wiley,NewYork,
2. H. Pollard,TheTheoryofAlgebraicNumbers,
1950.
THE U.S.A. MATHEMATICAI, OLYMPIAD
Sponsoredby theMathematicalAssociationof America,thefirstU.S.A. Mathematical Olympiadwas held on May 9, 1972. One hundredstudentsparticipated.
The eighttop rankingstudentswere:JamesSaxe, Albany,N.Y.; Thomas Hemphill,
Sepulveda, Calif.; David Vanderbilt,Garden City,N.Y.; Paul Harrington,
Central
Square, N.Y.; ArthurRubin,West Lafayette,Ind.; David Anick,New Shrewsbury,
N.J.; StevenRaher, Sioux City,Iowa; JamesShearer,Livermore,Calif. A detailed
reporton the Olympiadincludingthe problemsand solutionswill appear in the
March 1973 issue of the AmericanMathematicalMonthly.
The second U.S.A. MathematicalOlympiadwill be administeredon Tuesday,
May 1, 1973. Participationis by invitationonly. For furtherparticulars,please
contact the Chairman of the U.S.A. Mathematical Olympiad Committee,Dr.
Samuel L. Greitzer,MathematicsDepartment,Room 212, Smith Hall, Rutgers
University,Newark, N.J. 07102.
GLOSSARY
KATHARINE O'BRIEN, Portland,Maine
Campus disorder
Skew quadrilateral-quadrangle
extremepolarization
demonstration.
Generationgap
Two-parameter
familynegativeorientation
to communication.
Heart transplant
Removablediscontiniuity
binarycorrelation
operation.
Nylontires
Syntheticsubstitution
translationby rotation
transportation.
Popcorn
Iteratedkernelsonto magnification
transformation.
Suburb
Deleted neighborhoodlittleinclination
to integration.