Arithmetic of Ova Citation Clifford, Alfred Hoblitzelle (1933) Arithmetic of Ova. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/D6DV-QT27. https://resolver.caltech.edu/CaltechTHESIS:04082010-112340793 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Mathematics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1933 Record Number: CaltechTHESIS:04082010-112340793 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04082010-112340793 DOI: 10.7907/D6DV-QT27 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5709 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 08 Apr 2010 22:18 Last Modified: 03 Oct 2019 23:46 Thesis Files Preview PDF - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

Arithmetic of Ova Thesis by A. H. Clifford In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1933. le 26 3s He 5s INDEX Introduction. ccccccveccccccvscccseccsessese 1 Criteria for unique factorization in ova... Ideals and their fundamental properties.... 17 Decomposition of idealsSecccecvececssesceces 32 Ideal theory for regular OVaAccceccccceccece 37 Criteria for unique decomposition in terms Of LdealS.csceceece 48 Introduction. The property of unique decomposition into primes is fundamental in multiplicative arithmetic. The purpose of the present undertaking is to give criteria for this property, and to restore it by means of ideals when it is lacking, using only the single operation of multiplication. We start, therefore, with a system closed under a single binary operation, assumed to be associative and commutative. Following Belit:, we shall call such a system an ovum. An ovum is said to be regular if cancellation is permissable. A regular ovum is a commutative semi-group, as Dickson has defined this term. Criteria for unique decomposition in regular ova have been Biven by Koenig”* in very beautiful form indeed (Theorem 4.1). Conditions have likewise been given by Klein-Barmen®’ and by Wara*", the latter for the non-commutative case. We give criteria in § 1 for ova which are not necessarily regular (Theorem 1.1), and another set in §5 generalizing Koenig's result (Theorem 5.3). The former is applied to general conmutative rings, the result (Theorem 1.5) depending on a clever manipulation devised by Fraenkel” in order to obtain wnique decomposition in essentially finite rings. The concept of an ideal, as introduced in § 2, is essen- tially that due to Prufer®:. The definition is framed differently, however, in order to preserve the analogy with Dedekind ideals. In § 3 we postulate the Teilerkettensatz and the condition that every prime ideal by irreducible, and obtain the unique decom- position of any ideal into the product of mutually coprime, "“einartig", primary ideals. In § 4 we obtain criteria that the ideals in 4 regular ovum admit wnique decomposition into prime ideals, these being entirely analogous to those obtained by Noether’*. ‘The development in both §3 and § 4 follows van der Waerden®*, who follows Krul1?>, In 85 we note that, when we pass to rings, every ovoid ideal is also a ring ideal, but not always conversely. The two systems are multiply isomorphic, yet have in general very differ- ent arithmetic properties. It is noteworthy that if unique decom- position holds in the ring, then every ovoid ideal is a principal ideal, which is not necessarily the case for ring ideals. Thus ovoid ideals appear to have a more intimate connection with the multiplicative properties of the ring than do ring ideals, although they are not presumed to have such interesting additive properties, nor to be so useful in the study of algebraic mani- folds. Before passing on to the detailed development of the theory, I wish here to express my thanks to Professors E. T. Bell and M. Ward for many helpful suggestions in this endeavor. on tiie . ty T. Bell. “Unique decomposition". Amer. Math. Monthly 37, 1930, 400-418. Koenig. “Algebraischen Gréssen", Ch I. Leipzig, 1903. Klein-Barmen. "Uber gekoppelte Axiomensysteme in der Theorie der abstrakten Verlnupfungen". Math. Zeit. 37, 1933, 39-60. Ward. "“Postulates for an abstract arithmetic”. Proc. Natl. Acad. Sci. 14, 1928, 907-911. Fraenkel. "ber die Teiler der Null und die Zerlegung von Ringen". Jour. flr Math. 145, 1915, 139-176. Prufer. "Untersuchungen tiber Teilbarkeitseigenshaften in Korpern". Jour. fur Math. 168, 1932, 1-36. Noether. "“Abstrakter Aufbau der Idealtheorie in algebraischen Zahi- und Funktimenkdrperm". Math, Ann. 96, 1926, 26-61. L. van der Waerden. "Moderne Algebra” v 2, p 50 and p 97. Berlin, 19381. Krull. "Zur Theorie der allgemeinen Zahlringe." Math. Ann. 99, 1928, 51-70. l. Criteria for unique factorization in ova. A system consisting of a class S > an equality rela- tion = , and a binary operation o , will be called an ovum if the following postulates are satisfied: Py: To every pair @, 4 of elements of S there corresponds an element ¢@ unique to within equal elements. We write aeé=e Po: If @=a!’ and &=6' | then aeb=a'o&, P3: For every triplet 4, & », © of elements of § acl(hee) =(act)eoc. Pq: ack = boa, P5: There exists an element 2 in S such that Ro~ az 2oAarxra forall a ind. The element z of 2 is evidently unique. It will be ealled the identity element of & . The element qo & is called the product of a and & : and a and & are factors thereof. We shall write simply a¢ in place of “0 & . The inmediate consequences of (5 and “y are well-known’. Briefly, we may form the product of any finite number of elements of S . This product is independent of the order in which the * See van der Waerden, "Moderne Algebra", v 1, pp 20-22. successive products are formed, and of the order in which the factors occur. It depends only on the factors occurring there- in, and the frequency with which they occur. Powers of any element @ of 3S are defined in the usual way: Then the index laws ana” - Qt” (am™)” =a~™ hold for all non-negative integers 4%, %. An element a of S is said to divide, or to be a divisor of, an element & of S , if the equation axv= & has a solution x in S . G is then called a multiple of @. We indicate this by alé - The relation thus defined is transi- tive and reflexive, but not in general syrmetric. Divisors of the identity element z of S are called unities. They form a group. If al€ and bla always implies a= ¢& » Wwe shall say thet the owum S is reduced. tral ana & La then we write @u¢@, and say that a and & are associate. This is an equivalence relation with the property that if qua‘ ana & then ae ba'eh Hence oO forms a reduced ovum with respect to the binary operation © and » as the equality relation. This ovum we called the reduced ovum of o ‘ In the remainder of this section we consider only re- duced ova. The results hold for any ova provided we interpret ‘ Qa2#ag " as meaning " a is associate to i ," and *unigque™ as meaning “unique to within associates." Let, then, O be a reduced ovum. If ale > but a¢e » we write all € » and say that « is a proper divisor of i, This relation is readily seen to be transitive. An element of S which has no proper divisors othen than z is called irreducible; otherwise, reducible. An element « of & is called deconposable if proper divisors &e and ¢ of @ exist such that ~<=¢c., Otherwise, @ is called indecomposable. If a is indecomposable, and 4 =6¢ ; then either a= € or a=e 3; thatis, if c/a then a=axr. Henee the proper divisors of @ form a subovum of &. An element f of <S is called prime if the relation Vi la & implies that either fp? /a or fp? /¢ , and completely prime if - /aG implies that either pla or lia *;e being any positive integer. Every prime element is plainly inde- composable., Consider the sequence of elements « , as ; a g eeeey where & is any element of S . if they are not all distinct, let q* be the first element of the sequence which is equal to an element @ +A (A >o) further out in the sequence: a AraA a = A Ara . Lei | fer} A Now 4 a » so that « a . Likewise Arti a’ 1a . Since we are assuming that SS is reduced, this implies that We arrive in this way at the following result: If @ is any element of a reduced ovun S ; then either every element of the sequence & , an > a‘ y eee is distinct, or else they are all distinct up to a certain point, and all eaual from that point on. The number of distinct elements in the sequence will be called the index of @ 3; if all the powers of & are distinct, we shall say that @ is of infinite index. An element @ of &S is said to be decomposable into irreducible elements if distinct irreducibles (' pte Vike exist such that Me &% Xe a= Py (ve pa 2 where the x; are positive integers. The decomposition of a is said to be unique if the existence of another unique 3 3 bs a oat a of , 8 implies that (i) the sets br oe fr f , dg, ae ze are identical, so that, by suitable numeration, /=% oo and prrge > | 22/,8,4,2)3 (44) (i * if » @=1,2,.., 4), The second of these is equivalent to the statement that either a, = bs or else neither o, nor Es is less then the index of (i . | Theorem 1.1 The following conditions are necessary and sufficient that every element of a reduced ovum OS admit uniaque decomposi- tion into irreducible elements of oo : I. Teilerkettensatz: If a sequence 4, , @, , ... of elements of <S is such that ky, if G; » then the sequence terminates. II. very reducible element is decomposable. III. avery irreducible element is completely prime. Proof of Sufficiency: We show first that every element of SS is decomposable into a product of irreducible elements of So . For if an element 4 of S had not this property, then it could not be itself irreducible. Hence by II we could write Q= be ; where bil a ; cla . If both b and «@ were decomposable into irreducibles, then clearly «a would also be. Selecting the one which is not decomposable, we proceed as with a , obtaining a further proper divisor thereof not decompos- able into irreducibles. But this process gives rise to an unending sequence of proper divisors, in contradiction to I. Suppose now that we have two decompositions of @ : % a, _ &, (3 Since the result is evident for @=z we can assume a # z ’ and that the irreducibles /?, ,... , (”* are all different from 2 and from each other, and similarly for pe Meg a i / Bb, Ga , : Since (? , i ou i 4 we infer from an obvious exten- sion of III that i divides one of the $i , say i . This implies that P) = fi . Continuing with (feo Vike we get the result that J=% , and, by suitable numeration, Pr Ke > C= /, Ad. Since now fr divides none of the elements ¢,,..., i > cm it follows from III that it cannot divide peo fu . But x, | ora Se br ) ig i be “ br ? and hence from III again, i: / 4 . Similarly, iA fp) > and since & is reduced, > Similarly, Ss a ul Rs mo il cay v 7 vr QE 10 Proof of Necessity: Assuming now that every element of rs is uniquely de- composable into irreducible elements, we note first that if xy i Xr a = "i a f > (a; Pol then the divisors of a are the elements C Ca pow py where C, ranges from 0 to «,; . But these are finite in num- ber. Hence I is certainly true. let now a be any reducible element of “ . Let x, 4 a= pub, Gy rod, Then ~& =r and ¢ “op ge je are both proper divisors of @ , and 4>&¢ . Hence II is established. If now P [ab , where ” is irreducible, and if (? does not occur in the decomposition of 4 , then it must oceur in that of ¢ with multiplicity > %. Hence either Va Je or ple . This proves III. q.e.d. Theorem 1.2 The conditions I, II, III of Theorem 1.1 are independent. Independence of Tf: Take for example the set of all numbers of the form 27”, where a ranges over all non-negative real numbers, and 1 over all non-negative rational integers. Multiplication is the usual il variety: i (Zea) (20a) = 2 The only irreducibles are / and 7. Since they are evidently completely prime, ITI holds. Evidently II also holds. Yet I iL does not, e.g. the sequence <, Z _ Zz a Z*.,, violates it. g.e.d. Independence of ITI. Take for example the set of divisors of /< , the pro- duct of 4 and & being the L.C.Mof 4 ana &. Since the set is finite, I holds. The irreducibles are < and 3 , which are plainly prime (and hence completely prime, since the system is idempotent). Thus III holds. Yet II does not. For example, the only decompositions of 4 ere Y= 2° 4 ana ¥= Ho 4 , so that ~/ is indecomposable. g.e.d. Independence of III. Take for example the ovum io, z, aa A $ with multiplication table: (a [2 fv’ oO “5 (e) b oO A re) aA oO Oo az Since the set is finite, I holds. Since 7 is the only reducible element, and since 0 “/% ; Mi feo , f #O , it is decom- posable. Hence II holds. III does not, however, for ia “A? ’ so that 2 4 A, and yet f? ‘4 and (lr are both false. q.e.d. 12 If an element a of an ovum “5 has the property that AX = “7 always. implies + = qd » it is called regular; otherwise, irregular. Every regular element is clearly of in- finite index. The product of two regular elements is regular, and every divisor of a regular element is regular. An owm is said to be regular if all its elements are regular. If an element Z of an ovum eS has the property that Ax > # forall a« in ae » we call it a zero element. It is plainly unique, provided it exists. If 2 is decomposable: z= aq¢6 , #4 a ,2z4 & , then qa and are called nilfactors. Any nilfactor is clearly irregular, though, wnlike the case in rings, an irregular element is not necessarily a nilfactor. We have excluded the existence of a zero element in regular ova since it contributes nothing to the arithmetic theory thereof. Parenthetically, if a reduced ovum & has a zero # , and every element of JS » including 2 , admits unique decom- position into irreducibles, then ne is finite. In fact, if B= Brn pe then Oo is identical with the set of all elements of the form G oA "i ‘ fa as each C3 ranges independently from O to <;, So is therefore simply isomorphic with the reduced ovum of the x, ~~ XA residue class ring of any positive integer of the form q, oo, where 77 M, are any distinct prime numbers. a ne) 13 In a regular ovum, every reducible element is decom- posable. Likewise every prime element is completely prime. Hence from Theorem 1.1 we have immediately: Theorem 1.3 Necessary and sufficient conditions that every element of a regular ovum be uniquely decomposable into irreducibles are I. Teilerkettensatz. It. Every irreducible element is prime. If a and 4 are elements of AS », and if ¢ isa common divisor thereof such that every other common divisor of QA and b divides C , then C is said to be a greatest common divisor (G.C.D.) of « and &. If it exists, it is unique to within associates. Koenig showed that if every pair of elements of a reguiar ovum possess a G.C.D., then every irreducible element is prime, and thence to the property of unique factorization, by Theorem 1.3. Since this condition is evidently also necessary, we have Koenig's elegant result: Theorem 1.4 Necessary and sufficient conditions that every element of a regular ovum be uniquely decomposable into irreducibles are I. Teilerkettensatz. II. Existence of G.C.D. of every pair of elements. 14 Off hand we should suppose that similar criteria would hold for en irregular ovum, adjoining to these, say, the condi- tion that every reducible element be decomposable. Such is not the case, as the third example iy the proof of Theorem 1.2 shows. There every pair of elements has a G.C.D., and yet the prime property breaks down. The reason for this, and the proper gener- alization of Theorem 1.4 for irregular ova, will be given in § 5 (Theorem 5.3). We shall close this section with apvlications of Theorem 1.1 to general commutative rings having a principal indentity. The next theorem, giving criteria for general rings, is very Similar to a result of Fraenkel's mentioned in the Introduction. That the result is not true for ova can be seen by referring to the second example in the proof of Theorem 1.2. We remark that Since we have to use addition and subtraction we are not at liberty to pass to the reduced ovum. Theorem 1.5 Necessary and sufficient conditions that every element of a commutative ring R be uniquely decomposable into irreduc- ibles are: I. Teilerkettensatz: If ay, | a, » then the sequence 4a, , By yen terminates. 15 II. Vielfachenkettensatz: If @, il Gr,, , then either the sequence @,,4,).., terminates, or ‘no element # O is divisible by all the Ay - III. Every irreducible element is completely prime. Proof: The condition II is plainly necessary. Hence by Theorem 1.1 we have only to show that if these three conditions are satisfied, then every reducible element is decomposable. Let & be any reducible element of A . If @ is regular then it is decomposable. For if azbhe ,6Ha ,iW& , ale . tnen ables, aé/a, bli, 2 in contradiction to z | & . Hence a/¢ is false, and c/a. Now let a be irregular. By I every element has an irreducible divisor. Let p be an irreducible divisor of a . If @ were indecomposable then a=salf where QAua’, Hence 4 fp i a'p. so that anap a pics bee If now Via were regular, then & would be divisible by every member of the sequence pe Pope in contradiction to ITI. 16 Hence p is irregular. Since we are in a ring, (? is a divisor of zero, so that CF QO exists in A such that P es ° . Tet Gi = até . We proceed to show that a= (a'r Dp affords a decomposition of @ , i.e. that (aly ¢ ) Ma. For if @ /(a'ee) , then, since ala’, ale . let CGE: Then C= ape =@ tpe = tp Zz =O. But ¢2@#O - Hence Q@ cannot divide a’+e , so that (a’re ha . g.e.d. It is well known that if every ideal in a ring be a principal ideal, then I holds. We can also show that III holds. if K be a residue class ring of an algebraic ring A then (i) sinee every ideal in iat has a two-term basis, one of which can be chosen arbitrarily, it follows that every ideal in A is a principal ideal; (ii) since IT holds in A it holds in AC also. This yields a direct proof of the unique decomposition property in any residue class ring of a ring of algebraic integers. 1? 2. Ideals and their fundamental properties, If A and 6 are subclasses of an ovum BY , we shall denote the class sum and common part of 4 and 18 by A KB and A. b ; respectively. If A is a subclass of A we write A Cc b or /3 2 Al . We shall denote by AB the class of all products a vA », a8 aA Yranges over A and & over ah . If A consists of the single element 4 of. «5 » we may write a 6 in place of AB » The class of all common divisors of A will be denoted by L(A) . A subclass A of S will be called an ideal class, or simply an ideal, if it includes every element “J of a which has the property that J/¢ is divisible by all common divisors of the set Ax ; for every (fixed) element + of S . Expressed otherwise, if the class B has the property that, for each XX in S ; A (Bx) 2A(Ax), A 2B. then Whenever the letter X occurs in the following, the statement is assumed to hold for every %° in S - Ideals will be denoted by small German letters. let #/ be an ideal. Then if a class A has the property Ai«A) = Alxz) we say that A. generates, or is a generator of, the ideal a , Evidently every possible generator of Z isa subclass of Z . A Finite sé nerator fs G@aljed a basis. 18 Conversely, if A is any class, then the class 7 of all elements 4 with the property that 4x is divisible by every common divisor of A x , is an ideal, and A is a generator thereof. For then Afax) = A(Ax) and, from the way in which Z is defined, if then A 2 B . If A generates the ideal Z , we write Z=(A). If A consists of the elements 4 , &- s eee OF S » we IU write z=-(4,6,.. ). vd Theorem 2.1 A necessary and sufficient condition that (A) € (B) is that A(Axv)2A(Bx) for all * in oO. A necessary and sufficient condition that (A) = (&) is that A(Ax) A(x) tora xin S. Proof: il tet a =(A), A = (8), so that 19 AlAx) = A lax) A(Bx) = Alax), then we have to show that a necessary and sufficient condition that Hos is that A(zx) 2 ACbx). If gc @ then plainly Be & Ax so that Mae) 2 AC Ax). The converse is an immediate consequence of the defini- tion thet #7 be an ideal. The second part of the theorem is an obvious consequence of the first. , - ged. Theorem 2.2 it (A)¢(A’)ana (B) < (B’), then (AB) ¢ (AB). 12 (A ) =(A‘)ena (8) =( B'), then (AB) = (A'B'). Proof: By Theorem 2.1, A(Be) 2 Abe). Replacing % by ya in the first of these, we have ANAbx) 2 A(A'bx). NJ 20 Since this is true (for each fixed x in Oo ) for all & in 4), endhence all & in 6 , we infer that A\ABx) 2 ACA Bx). Again using Theorem 2.1, this gives us (AB) © (ABD). In similar fashion we prove that (A'B) © (ABI, and hence that (AB) €CAW). The second part of the theorem is an obvious consequence of the first. | q.6.d,. If g = yi and h = 1S are ideals, then the product of @ and {| is defined to be the ideal generated by 4G. We shall write this qZ- A or 7 fa , and in order to avoid confusion with the product of two classes, as already defined, we shall employ the convention that the juxtaposition of capital Latin letters will always denote simple class product, while the juxtaposition of small German letters will always denote ideal product. Thus az Lh = ( A Pe) ). Theorem 2.3 If A and (3 are any two subclasses of S » then (A). (8) =(AB)D. 21 Proof: Let a-A = (A) h = 8 = (8). Then by definition (A):(B) = 24 = (AB). (B ) = (3), so that by Theorem 2.2, (AB) = CA'BY, (A )(B) = (AB). q.e.d. Theorem 2.4 If g@ and la) are any ideals in 3S , then ah =ha. Proof: vet 2->A ,h=& . men since 48=64 we have (AB) = CBA). Hence by Theorem 2.3, (A)CB) = CBA). q.e.d. Theorem 2.5 ir Z, 6 , © are any ideals in oS . then ge Ob = a Aec. Proof: let g= A 4-6 sz. Then, since A-BC=Ab°C, 22 we have (A + 80) =(AB-C). Hence by Theorem 2.3, (A 080) = AB)(C) and again (A) (BNC) = (AX B) (0), q.e.d. The proof of Theorem 2.5 shows incidentally that aac = (ABC). of The extension to any number, factors is obvious. Theorem 2.6 teacsa’ ,A¢k’ , wen AAS aH Proof: Let a-A ; a’: A’ : 4-0 ; A’ 2B Then by hypoth- esis (A) < (AD (B)< Cp’). Hence by Theorem 2.2, (AB) ¢ (A'B') and by Theorem 2.3, (A )(B) Cc (A')C 8’). q.e.d. If ZZ A we say that the ideal qd divides, or is a divisor of, the ideal 4 ». faz oOo fe) » we say that qd is a proper divisor of bh . If g divides Vi » we say also that h is a multiple of Z. Bo An ideal £ is called a greatest common divisor (G.C.D.) of the ideals g@ ana A if (i) it is a common divisor of qd and A » end (ii) every common divisor of J and 4 divides € , An ideal £ is called a least common multiple (L.C.M.) of the ideals 7 and A if (i). it is a common multiple of q and Ht > (ii) every common multiple of 7 and A is a multiple of Lf . If Z and -4 have a G.C.D. it is plainly uniquely determined; we denote it by (Zz, A ) . Similarly for the L.C.M., which, if it exists, is denoted by la, & f. Theorem 2.7 Every pair of ideals @ , A possesses a G.C.D. In fact, ic g = (A), &=(B) , then (24,4) = (A“B), Proof: Let C _ A’ B c = (C) = (AB). We need only show that £ has the desired properties. Since C 2 A if follows that CC ) 2 (A ), that is, COd. Similarly, £ 2 fa . If now d2a2, 124 and d-D, then D> so that LD2C. Hence (D) 2¢c), that is, d q.e.d. tv) nh Theorem 2.8 Every pair of ideals Z ’ A possesses an L.C.M. In fact if Z =A ; 4 = io) ; then Proof: Let C =A, &. ite proceed to show that C is en ideal with the desired properties. To show the first, let the class D be such that Now CCcA so that Ce € Axe and A(Cx) 2 AA x), Similerly, A (Cx«)2ACBx). ence Ax.) 2 AA) and A (Du) 2ABx)- But 4 and %5 are ideals, so that DCA Dc 6, DEC. and hence Zo C is therefore an ideal. Let L= C Now CoA, CCG, i.e. cez, FEF d, Conversely, if dca ; dcfh on d= D, then DcecA, DE&b so that DEC, That is, d £&ec. q.e.d. Theorem 2.8 shows that the common part of two ideals is also an ideal, and indeed the L.C.M. a,A ever, in general an ideal. We thus sometimes write for da, Ad - The class sum of two ideals is not, how- Theorem 2.7 shows that if A generates @ and b generates A , then A - B generates (z ? A ) write (A, B) for (AYB) » so that (A, 8) +=4,4). We remark that (7, A)=(4, z ’ la, J- [Aa J, and also that if (z2,b) +84 or la, hli-a h2a « We shall then » and conversely. Theorem 2.9 Ir a ’ & ; L sare any ideals in S then (2, (4,c¢))= ((7, 4), 0) [a, Lo cl] lla, hic ld. 26 Proof: By Theorem 2.7, both members of the first equation have the common value ( A “2B “Ce D) ,» and both members of the second nave the common value A, 6. C. qed. we write (7,4,£) or (A,&,C) tor the first, and ld, hy, cl or a, 4, £ for the second, and similarly for any number off ideals. Theorem 2.10 Proof: We shall prove this for %=<X , the induction to any % being quite obvious. Let z-A, A= (3, La C. Evidently, 4 26 VC - AB VAC. The ideal generated by the class on the left is, by Theorem 2.3, (AB v2 ) » that is, by Theorem 2.7, a(4,c). The ideal generated by the class on the right is, by ; Lo Theorem 2.7, (CAB), CACY ,» that is, by Theorem 2.8, (2h,ac). Since these must be identical, q.e.d. The class S is evidently an ideal, which we denote by i , and call the unit ideal. 27 Theorem 2.11 If Z is any ideal, then TH = Z, Proof: A 2 If « is in and J is in <) » then Wax is divisible by every member of Z\ (A x) ,» so that 4@ is in A . Hence SA Cc A . 5 Since S contains an identity element, SA 2 A . Hence SA = A , and by Theorem 2.3, @7@a@ =a. qed. We remark that & 2 ah forall d@ . For S2A ; so that (8) Pp) (A) ; for all A . Theorem 2.12 a> ah. [a,4)2 a4. a2a rt 24, Proof: Hence by Theorems 2.6 and 2.11, Z 2 ah. Similarly h 2 ah. Hence d , Hh 2a th. g.e.d. If an ideal @ admits a basis A which consists of a Single element @ of S , then g is called a principal ideal. We write Z= (a). Theorem 2.13 The set of principal ideals en & is an ovum simply isomorphic with the reduced ovum of SS’ , Proof: To every @ in oe we let correspond the principal ideal (a) «- To every principal ideal Z we let correspond any element of & generating it. If (a) =l¢) then al & and tla » SO that a@& ke, Conversely if Aw Gy ,. then every divisor of <x is also a divisor of éx 4, and vice versa, so that (a) = (&), Hence the correspondence is C4; /) between the set of principal ideals in & and the reduced ovum of S . Moreover it is an isomorphism, since, by Theorem 2.3, (ab) lace). q.e.d. Theorem 2.14 tf (@,4)-o7 andl(z#,£)=c , then (ag, hc )= d. On multiplying and using Theorem 2.10, we obtain (a* di, ¢¢,4ce)-ord=:d. Now (7,4, 0) -<(r,c£)-a. On multiplying by @ , we obtain (44, 2b,dt)= a7 = da. By Theorem 2.9, then, (q%aG,4£, AL ) = (a, ac), whence (2, be) _ | q.e.d. 29 Theorem 2.15 Proof: By Theorem 2.10, [ah)-la,b)+ (labia, la,4]a), Now [a,hicd so that ld, blacd&. Similarly, la,hloe¢aa. wence = dy (dH) € al. qee.d. Theorem 2.16 if (a, tt) ={f , then la,4] = daly, Proof: From Theorem 2.12 we have la,4/2 24. From Theorem 2.15 we have la,41¢ a4. Hence la, al = dae. q.e.d. If (2, Aj)= , then @ and ff have no common divisor other than [7 , and we say that they are coprime. Theorem 2.17 If ad,,@,,.., dn are coprime$ in pairs, then Cit), days, an J = 4,d,°'‘d,, - Proof: By Theorem 2.16 the result is true for » =X . Assuming 30 it to be true for ~—/ > [ a, dy, dy, J = 4,4, a, By hypothesis, (72, a,) = & Clzri,n,, ue), Hence by an obvious extension of Theorem 2.14 for the case cl = LF, (7a. a, >a, )* a. By Theorem 2.16, then [d,, Leger, In |= L Lappin dn-, 4, an | ~ = Lad d,., > d.. | => d,d,' dn: Hence the result follows by induction. q.eed. Theorem 2.18 wt (7,h)=a ena @2GL , then dg QL. Proof: Let (q@,L/ 24, Then, by Theorem 2.14, (a, hey 2d. By hypothesis, Z2HL , so that gd = . Hence ae2c. q.e.d. An ideal having no proper divisor other than & is called irreducible. An ideal zi with the property that fi 2af implies that fi2 “or A 2 is callea prime. 31 Theorem 2.19 Every irreducible ideal is prime. Proof: Let A be irreducible, and let P 2 a@. Let (f,a) 2 b. Then £ 2 {7 , so thet either C=f or L=-gf . If fC+=A » then 2a . if g=z“4r » then Y 2G by Theorem 2.18. q.e oGle Theorem 2.20 A sufficient condition that every ideal in oo have a finite basis, is that the Teilerkettensatz hold for ideals in a , that is, if d,;,, > 4; , then the sequence Z,, Z,,... must terminate. Proof: let @, be any element of the ideal Z . If a= Ca) then the result is true. If 2a +7 (a,) , then there exists a, in Z but not in (a,) , so that If qd # (4a,, a, ) » then we can find a, in da but mot in (4,,%,) , 80 that Since this process gives rise to a sequence of proper divisors, it must terminate after a finite number of steps. 32 Hence we obtain a set of elements @,, 4, ,°+, “x of <S such that a4 = Ca,y t2 yy 64D an J. q.e.d. Unlike the case in rings, the Teilerkettensatz is not a necessary condition that every ideal have a finite basis. This is shown by the first example in the proof of Theorem 1.2. In fact here every ideal is a principal ideal. 5. Decomposition of ideals In this section we meke the following assumptions con- cerning the ideals of an ovum o I. Teilerkettensatz: If #:,, 2 #; , then the sequence Z,, @,,.-, terminates. II. Every prime ideal is irreducible. The final result to be obtained is the unique decomposition of every ideal into the product of mutually coprime, primary ideals (Theorem 3.4 ). Postulate II is the converse of Theorem 2.19, so that we speak indiscriminately of prime ideals and irreducible ideals. The following theorem depends only on the postulate I. Theorem 3.1 If Z is any ideal, then there exists a finite number of distinct prime ideals A, pict fla such that (i) Hh; 224 Ce el, 20, &D (ii) rd . hy’ , me fat ; («; >O Dd. iN 33 Proof: If @ is itself prime, the result is evident. If Z is not prime, then there must exist ideals h and £. such that y on Here Z2PH means that Z is not a divisor or H . Setting hi’ = (a,b) we note that 4’ > a; c'’>@Z . By Theorem 2.10, A's! =(a*,a4, #2, Ac), and, since @ is a common divisor of Z*, Zu,ac, He, Za 2 fic’. If both b and L’ have the desired property, then Z@ must have it. Hence if Z has it not, we can find a proper divisor of Z also not having it. Continuing thus we get an infinite sequence of proper divisors, in contradiction to I. q.e.d. An ideal q with the property that G2 ah always implies that y2 a. or else that q 2h° ,» for some positive integer © , will be called primary. Theorem 3.2 A primary ideal is characterized by the property that it has only. one prime ideal divisor other than ( . 34 Proof: let 4 be a primary ideal, and let distinct prime ideals Ai att fr be chosen as in Theorem 3.1 so that a ay Ho2 mp, Cp ze), for otherwise wn, ay A, 2 H.. ~ la 2 contrary to the prime property of P » the condition II, and the fact that Hence 4 must divide some power of Ar , say oy Oe, 4 >? f, But then Fs 2 4 > A, x, ; G re a) and hence P; = Ay We thus conclude that %=/ , and Ic the only prime divisor of 4 . Let now vi have only the single prime divisor A ~# a. Then by Theorem 3,1 an integer @ exists such that Suppose now that G2 au. it PPh , then (pf, 4/)= a vy 11, and hence by 35 Theorem 2.14, for the case d=/7, (Ho, b) =. Consequently ( g > A ) = oO and, by Theorem 2.18, 9 2aZ., If f 2h , then jw 2 be, and hence 4 2 At. Thus either 7 2 a or else vA 2 bh’, and 4 is therefore primary. q.e.d. Theorem 3.3 If HZ ana 4 have no prime divisor # 7 in common, then (2, f= 27. Proof: Let G@a,h)-= “L£. Then £ has no prime divisor other than @” , and hence by Theorem 3.1, 4£=@. q-e.d. Theorem 3.4 Every ideal is uniquely representable as the product of mutually coprime, primary ideals. Proof: let 7 be any ideal 4y7 , and let distinct prime ideals Prove Ja, be chosen as in Theorem 3.1 so that fi >a, (i = few a) Zz 2 po we ; Cx, po). Set Yd; _ (a ; A 2). 36 Clearly, F: is the only prime ideal divisor of D. other than @ , and hence by Theorem 3.3 (Hs gqd =e, Ce ey). Henee, by Theorem 2.17, [ay sens Gal = Ye Gar Gee On multiplying out the product Diet = GF a Bo) (a, BO) by means of Theorem 2.10, we see that each term in the resulting expression on the right is divisible by 2 But 9; a, (=t,..,4) so that 914, Dp 2@Z. Hence Zz. rc q,. tee Gn . Since by Theorem 3.2 each Ws is primary, this gives the « Hence WU h desired representation. Suppose now that we have another such : ¢ l ‘ gz 4, Y, Ge / ¢ Since A 2 5” 4, ' Ya it follows that A must divide one of the a. » say 5, . By Theorem 3.2, fh is the only prime ideal divisor of 9, » Hence ¢ a Re must divide one of the remaining Yj » say 4, » Continu- ing in this fashion, we get Jg=# , and, by proper numeration, P: 2 9: > (2 =/, ree, 2), 37 By Theorem 3.3, ya 2 2), Ce yi d> 7 2 (=k, Hence by Theorem 2.14, But 4 2 a Ga Ya Hence, by Theorem 2.18, i hv) Similarly we can show that ¢ J, = &j, + whence Hy 1! Likewise, 4 = Y. > C2 = Zi, a) and hence the representation is unique. q.e.d. 4, Ideal theory for regular ova. In the present section ay will denote a regular ovun,. a If we construct formally quotients z£ from o , and operate with them as with ordinary fractions, we obtain an abelian group, which we shall call the quotient-group of S » and which we shall denote by > « Since this is a well-known process we shall only very briefly give the steps involved. We say that 7 = z if and only if ad= fe, The relation = thus defined is found to be reflexive, symmetric, and transitive. 358 The product of two fractions is defined thus: 4.4 , 4 ¢ kL £4 We readily see that ir * = as , £. e! , then & 6. aH . a . “ a’ ce! € Lo Fk We then show that > is an abelian group with respect to = and « thus defined. The subset r , where 2 is the identity element of co, is simply isomorphic with Dy » and so may be “identified” with So by passing to ab whole new system isomorphic with S . The elements of 2. will be denoted by small Greek letters. An element & of 2 is said to divide an element 3 of D2 ,» relative to 25 , if the equation . ax= 8 has a solution x» in S . We then write « | 5 . This rela- tion is plainly transitive and reflexive. The class of common divisors of a subclass A of 2 will be denotea vty (4). 4 will ve called an © -moaule if it includes every subclass b of 2 having the property that A( es) 2 A(A¥) for every € in py - This is equivalent to saying that for every & in S . For if we put © = 7 and it Ad vein ACA Jinen xy is in AMAx) . Hence “4 isin J(Hxe)Jend Xd isin ABE). We can carry over the whole of the theory of ideals as presented in ge to & -modules. Since the set of ideals of oO is a subclass of the set of 3 -modules, it is permiss- able to multiply an ideal by an s -module. o “modules will be denoted by small German letters with a tilde, as Zz ’ unless otherwise designated. The purpose of the present section is to show that the following three conditions are necessary and sufficient that the system of ideals in oo constitute a regular arithmetic: I. Teilerkettensatz: If Z. ZHI > Z,; , then the sequence g,,d,,., must terminate. TI. Every prime ideal is irreducible. Iii. S is integrally closed in a e The meaning of III is that if x in 2 is such that a exists in S such that ag” is in oS for every positive integer > , then & isin &. Theorem 4.1 if © contains every element f of 2 with the property that pr Cwm , then zd =P , and conversely. Proof: Evidently cl(x/=A(@7</, so that by Theorem 2.1 it 40 suffices to show that A fax) © Ave). For then @g@ 2” , whence Z = . Let A be in A(ax) ; so that d lax for every « in Z . Set fr a . Then pa isin & for alla in as, that is, pe ¢ @& . Hence by hypothesis f is in S , so that d/xe , ana Alae) ¢ dhe). To prove the converse we note that if fer gS I, then Pe = FP isin @& . qed. Theorem 4,2 If @ be any ideal, then the class P of elements f — of a. such that P ac & is an § -module. Proof: Let & be any element of >. such that Sx is divi- sible by every member of A Px) . Then Sa is divisible by every member of A(Pa) , for each a@ in @ . Since all the members of Pa are in 3S , the element 2 occurs in Z\ (Pa). Hence Sa is divisible by z , that is, Sa is in § . Hence, by definition of P ’ § isin /? . P is consequently an S -module. q.e.d. 41 The © ~module defined by Theorem 4.2 for an ideal Z. , will be denoted by Z™’. Theorem 4.3 If 2 #4, then @~’ contains an element s of 2 not in S&S . Proof: If every element of a were in BS » then, by definition of ao ; & would contain every element f of 2, with the property that fe ¢ Mm . Hence by Theorem 4.1, 2 =@& , con- trary to hypothesis. q.ee.d. Theorem 4,4 For every prime ideal yi ; A “ose, Proof: Since Ao 2 we have 72H’ 2 Pp. Hence by II, Bf + fA or Be =o. tf AR” = pF , then -~1 Hence if A be any element of A » and any element of fi ; the elements ad" of 2 lie in B . Consequently, by III, is in & . Thus we conclude that every element of x’ is in es ; 42 in contradiction to Theorem 4.3. Hence FP" =a. q.e.d. Theorem 4.5 Every ideal is representable as the product of a finite number of prime ideals. Proof: let Z be any ideal. Then by Theorem 3.1 there exists a set fu, bi fa of prime ideal divisors of Zz , not all necessarily distinct, such that Zz 2 fi Pas fa Choose a set with this property, and such that @ is as small as possible. If £=/ , then F, 2a 2 ,» and @ >f,. Assume the result to be true for all ideals for which z-/ primes can be found with the above property. Then if @ require exactly A primes at least, we have fy 2&2 fife pa: Multiplying by A, and using Theorem 4.4, Ir 22a f,' 2 fact fae Hence a fa,’ is an ideal (since it is an © -module included in # ), admitting the result of Theorem 3.1 for %-/ primes. By hypothesis for induction 43 Multiplying by fa ; _ Ls 1s a fps ae q.e.d. Theorem 4.6 If a= But Bes b = Ao AL , and #2 qa , then every prime ideal # ff occurring in the representation of h occurs in that of Z7 , and in fact at least as often. The theorem is trivial for 4 =m . Hence we can assume A>o and by #e- Since hy 2h2 A,-+) Ax we have from the prime property of fA,” that 7, must divide one of the Be, say f . Using II, we infer that Multiplying the rehation Poo BF 2 yo fla =! by 7, , we get ; i. Ps ose fa. 2 peo an The theorem is evidently true for J7= / , as the result shows. Assuming it to be true for all products h of less than 4 primes, then fle ys Fy all occur among the primes fas» rey P> 44 repeated ones occurring at least as frequently. Hence the same result holds when we adjoin BY to the former set, and its equal B, to the latter. q.e.d. Combining Theorems 4.5 and 4.6 we have inmediately: Theorem 4.7 Every ideal is uniquely representable as the product of a finite number of primes, the multiplicity of each prime being uniquely determined. Theorem 4.8 if 2H=A2L ,then fF zl. [ An immediate consequence of Theorem 4.7.4 Theorem 4.9 If 42h , then C exists such that @e =f. Proof: Define [ to be the product of those primes in the re- presentation of b which are left after those in the representa-~ tion of @ have been removed. The result is then clear. q.e.d. Theorem 4.10 The conditions I, II, III for a regular ovum & are equivalent to the following conditions: IV. Hvery ideal in So is uniquely representable as the product of a finite number of prime ideals. V. If Z@ 2 é then # exists such that 2H =2. VI. If ZH=2C , then (=e. [this is a conse- 45 quence of IV if we assume that multiplicities are unicuely determined. | Proof: I follows on converting the usual Teilerkettensatz for an arithmetic to the Teilerkettensatz for ideals by means of V. II follows from the fact that [7 = 7 is the unique representation of a prime fA » 80 that it can have no proper divisor # {*. It remains now to show III. Let & be an element of > with the property that @ exists in S such that aa ™ is in 5 for every positive integer ~ . We are to show that x lies in JS . let @ be the ideal generated by the elements A, ak, AK*,... of 2) : a=: (4,4a%, a, SE dD Then a* > (a*, ee Using VI, A = la). . . @ Consequently 4 [aw , so that 2 / « , and « is in JS. q.e.d. 46 Theorem 4.11 If 2Z@ be any ideal, then an ideal A and a principal ideal (¢ ) exist such that ZH =e). Proof: We need only take ¢ to be an element of #& . Then 4 2(¢) ana the result follows from V. q.e.d. We shall conclude this section with a brief account of fractional ideals. By a fractional ideal Zz we mean an oc module for which & exists in aS such that Zz is an ideal. Theorem 4.12 Necessary and sufficient for Zz to be a fractional ideal is that Zz have a finite basis. Proof: If T= (dl), 005 Xn7 then by multiplying by the product 4 of the denominators of the «, we see that 6H Op. Conversely, if & a is an ideal, then by Th 2.20 and I it has a finite basis: ot a = Ca fottte uM A y) , Hence ° : qg.e.d. Theorem 4.13 If #Z is any ideal, then 2’ is a fractional ideal. Proof: If 4 is any element of Z, then ad’ ow. qeeed. 47 Theorem 4,14 — The set of fractional ideals in 7, forms an abelian group with respect to multiplication. Proof: 2 rakica, ¢h cr, then (a €)(F a Cir; hence the closure property. The associative and commutative laws and the existence of identity are clear. Let £ be any fractional ideal, so that -~ exists in such that @£ = 2 ¢ @& . By IV, By Theorem 4.13, hy! pitty Fa are fractional ideals. Since pat | we see that c has the inverse oy i ~1 aad C —_ fh eu [+e . Hence the set is an abelian group. q.e.d. We shall denote a 7! by a Theorem 4.15 The group of fractional ideals is the quotient-group co — of the ovum of ideals in oO . Every fractional ideal gd is representable in the form 2 b ~ Pave fe P. wc Ps Cy where H ana £ are coprime, and the sets tH, yen, fa tS , LBs 6s. , Be f have no element in common. Proof: Since ¢ exists in such that cga-omr we have that w= b c where L =(¢ ) cf, If now we represent & and 2 as the product of primes, and cancel those occurring in both numerator and denominator, we get the desired representation. q.e.d. 5. Criteria for unigue decomposition in terms of ideals. In this section we show that the condition that every irreducible element be completely prime, occurring in the criteria for unique decomposition (Theorem 1.1), can be replaced by the condition that every ideal be a principal ideal. The fact that, unlike the case of principal ideals in ring theory, this condition is necessary, points to the conclusion that ovoid ideals have a more intimate connection with the miltiplicative properties of the ovum or ring than do ring ideals. For example, in the ring Cle! of polynomials with integer coefficients every ovoid ideal is a principal ideal, whereas polynomial ideals are notor- iously lacking in the usual properties of an arithmetic. Ovoid ideals, however, are not presumed to have the interesting additive properties of ring ideals, nor the utility of polynomial ideals 49 in the study of algebraic manifolds. As a matter of fact, every ovoid ideal defined for a ring HK is also a ring ideal, though not conversely. Since aix - 2 , Avis in Z forevery 4 in @ and every & in 4 . If 4 and 2 are in d , theniu,ru,/e is divisible by every common divisor of the set d *« (since 4, and 4, ¢ are). Hence 4d is closed under addition and sub- traction, and under multiplication by any element of A . As an example of a ring ideal which is not an ovoid ideal we have (2, ~) in the ring City. Since f/x) is divisible by every common divisor of <jix) and ¥ fle), the element f should occur in (<, «) , which it does not. To every ring ideal there corresponds a unique ovoid ideal, nemely that generated by any generator of the ring ideal, including the ring ideal itself. As we have noted, the corres- pondence is many-one. It is, moreover, an isomorphism. Hence the arithmetic of ring ideals is multiply isomorphic to that of ovoid ideals. | Turning now to the matter of principal ideals in an ovum S » we ask, what is the significance of the equation (4,#) = (cd), 2 where 4, ¢ , ¢ are elements of © ? It must be remarked that it means decidedly more than that <¢ be the G. C.D. of « and &., It says that for every « in SS > Cx is the G.Cc.D. 50 - trae of 4x and +x, The same is ture for ring ideals, since then ¢ is a linear combination of q and & . Thus the above relation is in a sense a generalization to an ovum of the notion of a linear function in a ring. This is borne out by the way ovoid ideals multiply: (a, wdle, az) z= lac, WO ak, wk /, See also the remark after Theorem 5.2. If we replace the ideals in Theorem 2.14 by principal ideals, we obtain the following: Theorem 5.1 Te 1a,¢/= 12) and (u,6) - CX), then Theorem 5.2 If (4,¢/ = (¢) then « lee implies + le. Proof: If(u,*+/ = (Cé/) then every common divisor of 1 x and «x is a divisor of v . if alo’ , then a is a common divisor of .c and #< , whence 2 fe. q.6.d. This simple but fundamental result shows that (a, &/=(2/) is the proper generalization of the notion of "linearly coprime" elements, i.e. elements & and 4 such that tay 4 eel, 51 Theorem 5.3 Necessary and sufficient conditions for unique decom- position into irreducibles in a reduced ovum are: I. Teilerkettensatz (as in Theorem 1.1). Ii. Every reducible element is decomposable (as in Theorem 1.1). TIiI. Every ideal is a principal ideal. Proof of Sufficiency We have to show that III implies that every irreducible element be completely prime. Let P be an irreducible element, and let 2 is irreducible, either <= 4 or <2. { Since d / je and if A= p » then yo /a If 4=2 , then by a successive application of Theorem 5.1, (4, pr) = Ce). Hence by Theorem 5.2, ple. : 2 / ; & Gg ' Hence either /° & or - /e, so that i is completely prime. q.e@.d. 52 Proof of Necessity. let a2 and & be any two elements of S , and let "ld prey "ik be the irreducibles occurring in the decompos~ x itions of 4 and & - Let a 4 (Sa where some of the a; or oe may be zero, but where none exceeds i) S SS, the index of the corresponding (i ; The element c= phi. ph where Yr somen (ap, So) 5 Comte tb) 2 is evidently the G.C.D. of a and © . let + be an arbitrary element of S » and let ¥, Fy Then since it follows that ¢xv = f: tee i is the G.C.D. of X,- &, ‘ wat Va and nae tx = pr Hence (a ) &/) =(e) 53 lat now @ be any ideal. Let 4, be any element of mw . if a = (4, ) then the desired result is true. If not, then &, exists in A but not in G,) - let (a,)= la,, &d. Then 2, li a, , for if 4, = 4, then 4, (é, ; and £& would be in (a,). Evidently 4, is in Z.. If Z= (a,) then the desired result is true. If not, then 4, exists in such thet a, /4. . But I is a consequence of unique decomposition, so that this process must terminate, and Z = la. y) for some integer ~ , q.e.d. This theorem gives the correct generalization of Koenig's Theorem 1.4. It has already been noted that III is equivalent to the following two postulates: III,: Every pair a, & of elements of S has a G.C.D. (a, €). TII,: (a,é)e =Cac, Ge) for all <, €,<¢ «x S. The third example in the proof of Theorem 1.2 illustrates the necessity of III,. It satisfies I, II, and III, , but not ITI, . For 2 is the G.C.D. of ? and é , but x is not the G.C.D. of pe and fi for all x . For example, put x= fo : the G.C.D. of poze and ia Oi oO, not 7° . It should be remarked that, unlike the case in rings, I is not a consequence of III. fhe first example in the proof of Theorem 1.2 illustrates this. See also the remark at the end of § 2. Applying this result to that of Theorem 1.5 we get the following: Theorem 5.4 Necessary and sufficient conditions that every element of a commutative ring KR be uniquely decompos-— able into irreducibles are: I. MTedlerkettensatz (as in Theorem 1.5). II. Vielfachenkettensatz (as in Theorem 1.5). III. Every pair @, & of elements of A has a G.cav., (4, $), in. w. (4,4)ewlac, ce ) tor alla, &,cinKe