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 . Accessed: 08/11/2011 11:53 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to Mathematics Magazine. http://www.jstor.org 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.

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