The Lattice Structure of Algebraic Models Citation Harrison, Gerald (1943) The Lattice Structure of Algebraic Models. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/d2a9-xs49. https://resolver.caltech.edu/CaltechTHESIS:03252025-180453431 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): Ward, Morgan Thesis Committee: Unknown, Unknown Defense Date: 1 January 1943 Record Number: CaltechTHESIS:03252025-180453431 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:03252025-180453431 DOI: 10.7907/d2a9-xs49 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17090 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 31 Mar 2025 19:08 Last Modified: 31 Mar 2025 19:26 Thesis Files PDF - Final Version See Usage Policy. 8MB Repository Staff Only: item control page

The La ttice Structure of Al gebra ic Moduls Thesis by Gerald Harrison In Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technolo gy Pasadena, California 1943 The writer wishes to express his sincere appreciation for the encouragement and advice of Professor Morgan Ward, under whose direction this dissertation was written. Summary of Thesis The thesis consists of two parts. In Section I the theory of lattices \mose elements are mm matrices Yd th elements in a principal ideal ring is developed. Here the main result is a decompo s ition theorem stating that such lattices are a direct product of certain well distinguished sublattices whose elements a.re 11 primary11 elements. The results have an immediate application to the theory of linear form moduls over a principal ideal ring, and in fact the theory was developed with this in mind. Section II gives a rather remarkable pro perty of the above mentioned decomposition theorem when the basis elements of the linear form modul which the matrix represents are alt;ebra ic integers, and is therefore a contri"bution to the multiplicative theory of alt;ebraic moduls. Intro ducti on It was p robably Dedek ind• s study of algebraic moduls in developing the th eory of algebraic numbers which led him to introduce and. study the abstract al gebra. called by him a 11 Dualg rupp e 11 , now celled a lat t i c e. He showed t ha t the f ree moclu lar lattice generated by three elements is of order t wentyeight, a nd exhibited set s of quadra tic mo cluls which a ctually g enera te it (D edekind ( 2] ). However the modul a r lattice of submoduls of a lin ear form modul i s not the most g eneral discrete mo dular l a ttic e ; thus, thoug.~ the free mo dular lattice on f our g enerators is inf init e , the free lattice g enera ted by any f inite numb er of su bmodul s of a moclul of f inite ord.e r is finite. The ques t ion naturaJ.ly a ri se s as to what l a ttice i d entiti es oe s i des the mo dul a r i d entity are satisfi ed by t he lattice of submod.uls of a linea r form modul. Though this is as yet an unanswered question, the resu lt s of Section I si~lifies the p roblem. The study of al gebraic modul s may be d ivicled into t wo p a rts. In the first t h e t heory i s simpl y thc:.t of a linea r form E1odul over the ring of rational intege rs, i7hil e in the second the al gebra ic character of t h e b~s i s elements p l e.;y-s an es s ential role and leads to the theory of algebra ic numb ers and. i deals . Section I conta ins cont ributions to the former theory, and Section II to t h e l a tt e r . It was fo uncl_ tha t b oth theories could be treated most easily by employing matrices; indeed the first section mey be considered a study of the lattice theory of matrices with element s in a pr incipal i deal ring. Th~ usual -v and" notation of lattice theory is usecl (c f . Birkhoff ll1). If a,, I:, , C , • • • are elements of a lattice, a, covers h means (i) a.,>i, (ii) a~/J, and (iii) Q >c>b implies a :::e or c-== b . Arrows such as - - and --- will be used to indicate one ,vey and. two way implica tion, respectively. If A and. b be written by are elements of a ring, a divides aJ .6 {a.,.t>),::: / . b rrill sometimes and ct ,:nd D relatively prime will be incl.i cated Section I Let 1'! be a closed associative system with elements ,4, 13., C.) • and a unique right and left unit element /. The closure operation is written as a multiplication and the identity ,4 and l3 of 1/'( is indicated by A=E. of two elements The set consisting of the elements of ?ft forms a partiaJ.ly ordered set tl if the ordering relation > is defined by A> 13 -- -►- B =/IC,, and the associated equivalence relation ~ by 4,,..3 ---- - A>B.,8>4., For 4 == 41 implies A' >4 for ea.ch element A of JJ( , and the transitivity of the ordering relation may easily be verified to depend on the associativity of the closure operation of 7;;'J. Since A== J,4 it follows .f ;;,4 for each element ,4 of !) ; thus I, the unit element of ?!Z, is also the unit element of t) Of course there is another partially ordered set t}.,, which the elements of ?.q'-' form with respect to the ordering relation of right division, i.e. 4 1 B ----- B = C.4. However [) and our attention to ff are abstractly identical end we will confine tJ. The set of elements of ~ which possess inverses forms a group which will be denoted by !I. JI is not a null set since - 2 - J is its own inverse . I t is easily se en tha t A~/3 if and on.ly if the re is an el ement J of An element .1 such tha t 17 :::BJ. P of ?Jt will be cal.lea. a p ri me i f P ~48 i m- p lie s that one a nd only one of A or l3 is an el ement of Y. I t i s a simple matter to show that A covers B - 1-3 =,4 P, P a p rime . For A >f3 i mplies lJ =..4P , and A :t=B implies P is not an element of ,f . Supp ose P .:3 t,)R; then 8 ;:4QR, and 4:> A O>AQR =B. Sinc e 4 cove rs 23, either ,4-=A~ or A (} -:,,1/t;)f?; henc e one an:l only one of Q or R is in 1 , and P is a p rime . However the conve rse 8 :i AP (1.1) , Pa p rime - >- A covers f3 i s not in g eneral· true without some additional assumption on ?n If o==.1/P, P a p rime, it does follo w that ,4 >8 and 4~8 . Suppose 4 >C>B ; then B = C-R, C ::c AQ, and B :: 4()R. Thus B = AQR =' AP (1. 2) 1 If 'lft. is a semi- g roup in which left cancellation is p ermissible we may conclude QR= P, and since R is in §, thus p rovi ng (1.1). P i s a p rime, one of '9 or However left cancellat ion is only a suff ici ent cond ition for (1.1) to hol d , and we shall see in our study of matrice s tha t (1.1) mey be p roved by an a r gument on the 11 n orms 11 of the elements involved without any app ee.l to the cancellation law . I f P is a p rime and P=.?, then Q i s a p rime, i. e . P a p rime i mp lie s ea ch element of the set Pf is a p rime . P J!P is a p rime, thru gh is a p rime then eac..h. element of the set Also if - 3 - p ,,t JP in general, for J in .1 . a clas s of primes Thus there mey- be associated JP with each prime P of ?I/. and each prime of J,r is in one and only one of these classes. Suppose the following condition holds: ,; . If an el ement ;!/ of '1lt has one decomposition as the product of primes belonging to a single class JJP. Pa prime, then all decompositions of A as a product of primes has as its prime factors only primes of J'P. A will then be called a primary element with respect to 1'P. ~ implies the set of primary elements with resp ect to a fixed set of primes ,jp is closed under the closure operation of /;?, i.e. if 4 and l3 are primary elements with respect to then AB is also a primary element with respect to !P, JP. Thus if tP denotes the set of all primary elements with respect to .1P together with the set .sJ, then <Pis a closed associative system with the same unique right and left unit I as If. The elements of (}J aJ.so form a partially ordered set with respect to left division, which we denote by ~. We will now investigate an important instance-of the above theory in which (i) the partially ordered set~ is a lattice, (ii) each partially ordered set G6~ is a sublattice of J. and (ill) the lattice !) is a direct product of the sublnttices ~. -4- Henceforth 'J?t shall denote the set of all non-singular n,rn matrices with elements in the ring It of rational integers. The determinant of a matrix A will be called its norm and will be denoted by n(A) . if i I n r;4; ~ 0 . Thus a matrix A is non-singular if and only The associative closure operation is taken as row by column matric nru.ltiplication, and the unique right and left unit / is the n..r17 matrix with 1 1 s positions and 0 1 s elsewhere. in the principal diagonal Though we are restricting Ir to be the ring of rational integers in order to simplify the notation and avoid circumlocutions regarding associates, many of our main results hold_ when ~ is any principal ideal ring. Matrices which possess inverses, i.e. element s of the group j , will be called unimodular, while the term unit matrix will be reserved for I (cf. MacDuffee [l], p. 30). A matrix is unimodular if and only if its norm is a unit of ft (MacDu.ffee LlJ , Theorem 20.1). If A= BJ, where J is unimodular, we shall call A and 8 right associates or simply associates. Thus A=B if and only if A and 8 are associates. Matrices which are neither unimo d.ular nor prime will be called composite. A composite matrix can be expressed as a product of at most a finite rnunber of primes (MacDuffee (ll , Theorem 20.3). It is clear thGt if the norm of a matrix 4 is a prime of Or, then A is a prime of Jil.. corollary of the following theorem. The converse is a - 5 - Theorem 1.1. If the norm of a non-singular matrix A is divisible by a prime p of (R, there exists a prime matrix P of 7/! such that (i) n <P) :::p, Ci) A 'ii BP, 8 an element of i/1( Of course fi(A) =- 17(8 ) ? . In the p roof of this theorem we shall apply certain wellknown results concerning matrices in Hermite's normal form (see Ma cDuffee [l] , Theorem 22.1). Let A, be an an associ a te of A which is in Hermite 1 s norma l form. where J is unimo dular, and Then /4 is of the form wher e all elements below the p rincipal diagonal are O. Now nW,)-=- rX 1' 14'~' 2 • •• cx'v7 , and since p divides n (/4 J , p p rime, if follows p divides some 4-'~~i ; let «"·" be the 0( of smallest superscripts which is divisible by p. ~ ,,, /3 r,:L . - . f3"'" (?,'- - . (2,'l'"' Let '1 ,,,, ... ~ P, ~ 8 k,.,,,, p I %'" a ,I - 6 - where p occurs in the r' th row and r• th column of P,. Then f3r.r? \ Let Then 4, = B t:::, if integers k,j,. "k.,,., .... f3 , , f3 I',(' exist such that The first equation is solvable for /u~,·,p) "'( ~ ., k,...,,, and (3~" by hypothesis. Since ( f3't",' , p ) ~ 1 , l .-< r , the remaining equations may be solved one afte r the other, t he seco~d for ./r,._,_,. the third for .4,... , r for k,,r and f3 '," . and /3,...,,, , , and finally the r' th Hence - 7 - where P := P,J I? (4) == n4'3} p n(P):::p, as was to be shown. and That follows from the generaJ. theorem that if l3 are any IU/l matrices, then Corollary 1. 1. 4 and a/48) = n (A) n(BJ. An element P of Ji(_ is a prime of ll;' if and only if n (P) is a prime of rf?. If nfP) ~ p , we say that ,D belong s to p . The following theorem is obtained by an obvious induction on Theorem 1.1. Theorem 1. 2. Each element of 1')[ has a decomposition as a product of primes ,•.rhich belong to the prime factors of the norm of the given matrix. Theorem 1. 3. ✓4 covers 8 if and. only if B=4P) P a prime. If a(P) :- p we say that 4 covers 8 with respect to P . The suf f iciency has already been demonstrated in the general tlieory. In p roving the necessity assume we have arrived at equation (1.2) in the argument. Then rt f4 ) n (() l F? (/f) == n(4) 1/P) or /J/Q ) n(/1) which , ,i. th the aid of the corollary above and a prop erty of p rimes of (if , conp lete s the proof o f the theorem. - 8 - 4 covers 8 if and only if Coroll a ry 1. 3. (i) A>B> , P a prime of 6f . { ii ) 17 (8) ::: l'1 (,4) p The followin g t wo theorem s follow directl y from The orems 1.2 and 1.3. Theorem 1. 4 . Let A, B , C be non-singular matrices such that ci) 4 > B , B =A'C , {ii) !V(c ) -: : P,e'j{4. . • . Ptet , where p ,;?~ e.re p rimes of •• , ?~ O? . Then every p rime cha in joining .4 and contains ::fe. f-/ I J - B elements (both 4 and 8 are included in the chain). Theorem 1. 5 . A non-singular matrix has a s many distinct decompositions as a :9 roduct of p ri mes as there are different p rime chains joining the matrix to the unit matrix, where t wo decompositions are distinct i f not all corresponding p rime matric factors are equal, and two chains are different if t hey are no t identical. The following theorem holds only when (fr is the ring of r a tionaJ. integers. Theorem 1.6. There are p recisely p"- 1,/ p-/ non-associate - 9 - prime 11.(f'l matrices belonging to each p rime p of If. The prime matrices are "\ P k., .. •. • k,,., I I I I ) ' I ... , p I k-,"v· s t<_; ~ll, (/,,.1,:2.,. •·, nJ \'V'here each ranges independently through all integer values from D to p-1 . The proof is omitted since it is simply a matter of enUJnerating cases and applying the uniqueness of the canonical associ- P ate of a matrix (see MacDuffee (11, Corollary 22.2) If denotes any one of the prime matrices belonging top enumerated in the above theorem, then the set ..,g P of primes referred to in the general -theory is the set of primes given in the above theorem together with their associates, i.e . .ffp is the set of all primes belonging top. By an argument on the norms of the matrices involved it is clear that condition ~ is fulfillecl, and that a primary element of ?;1( is an element whose norm is a power,, say ,.~ / , of a p rime of c,1 . r will be called the rank of the primary matrix, and the primary matrix will be said to belong to the prime of {f( of which its norm is a power. primary element of rank 1. Thus a prime is a The rank of a primary matrh will - 10 - usually be written as a superscript in parenthesis, e.g. P Each pair of elements A and 8 of 6 irl . has a greatest commcn left divisor Au B and a least common left multiple A,., B . This was proved by Ohatelet 111 when cf is the ring of rational integers. However this can be proved most easily when df j_ s any rin g by ap9eaJ.inc to the theory of linear form moduls of order 11. over a ring rJ?. If ;tis such a linear form modul the sub- moduls bf o rd.er 11 of £ form a lattice, in fact a modular lattice, with respect to union and crosscut. If we think of the non-singular 17-trl matrices with elements in <Jc as representa.tives of the submoduls of of, we can immecliately conclude that these matrices form a modular latti.c e with respect t o the orde!'ing relation of left division. ( See MacDuff ee [l] , Theorem 24.1, Corollary 24.1, e~d Theorem 24.2). Thus associates exist when ana_ only when tR has units, and then they- represent the same submodul of ~. Before proceeding with the study of the lattice interesting to note that when f? tJ it is is a principal ideal ring the set of l?-tf/ matrices with elements in <Ji. satisfy the following five conditions with respect to matric multiplication: ( i) closure, (i~ as so cia ti vi ty, (lit) A -:: B - -a.- C!A -: CB , ( iv) Cl4vB) (v) =CA'vCB > I,4 -: ,4 I-:A 1 I the unit matrix. - 11- Thus matric multiplication is a non-commutative ideai multiplication over the lattice of Since l'IV'l matrices with elements in rfr. !} is a modular lat t ice, both Birkhoff covering cond itions are satisfied (Birkb.off 111, page 34). This fact is used in proving the next theorem. Theorem 1.7. If A and 8 are elements of !) such that A vB covers 8 with respect to the prime F' , then 4 covers 4,,,,8 with respect to p . By an inductive argument the proof mey be reduced to the case in which A u8 covers 4 cover A'18. Each of the s e coverings i s with resp ect to p rimes a s well as 8 , so that A and B of W, so that a : (Av B) P, Ari 13 ~ 8 O where ?, P : 0 , Q I A ~ U v B JP ', . A '""I B = AO 1 • are p rimes and we are given ,/P) =p . Since we have where ea ch of r;(P), n 1()) , n//)IJ , and n(Q'./ is a p rime of Or. Hence if /?({;) "' 9 we mey conclud e that either n (P ~,I-.::? and or 11( c? J-= p , - 12 - In the former case the theorem is proved. latter po s s ibility holds. Assume that the We will show that 1t follows that p= '!, completing the proof. In order to simplify the p roof we will appeaJ. to the c ancellation law, though this is not necessary. Then P Q ~ PbE C, say, 1 P and P are primes both of \vhich cover C, we have C-= p,,,-, p~ and since Let Pand P' be in Hermite's normal form, so that l ·p k,,:s,., where aJ.l omitted elements are O•s. P and P I If f;t:.S it is clear that can not be multiplied on the ri ght by prime matrices in Hermite's normal form to obtain matrices in Hermite's normal form havine the same norm unless p=9 . Q - 9 /,..,.,.., •• . l,,,, , Assume r-=s, and let , 'l l ,.,jr,., Q'-= I I Then I P't 7',., ~ p2,.,,.+l+ kr.r+• •• • • - 13 - 1 I Thus P <;) -== P Q if and only if Hence and 1 P=P , inr_plying Jf~B, contrary to our hypothesis that 4. c P'.J covers F'..> • Hence the theorem. An obvious induction gives the following result. Theorem 1.8. If A and 8 are elements of CJ, then r?f4} n (8 ) e n(4.,, B) n (4uB}. The norm is thus a multiplicative type of modular functional. If n t4):- r{'fl..-··PJ-~ the p, primes, the usuaJ. ad- ditive modular functional is p t4 ),,. $e.. , and l 4) +-(J(B) • p 1A r1B) +p(.4/ uB) • Theorem 1. 9. The set of all primary elements of 8 belonging to a fixed prime, sey p , together with the set sf of unimodular elements form a dense modular sublattice of&. lattice will be denoted by This sub- 0,. This follows directly from Theo rem 1. 8, since ,,,(4).., pr , .,z (B)=-p" implies 11 (A,,8) -=p4. , n(4t.:?.)"l: p ., , where N:s,,,uiv. We adopt the convention of caJ.ling a unimodular element a primary element of rank zero and belonging to each p:-j.me of (fj'. ~J Thus P denotes a unimodular matrix. - 14 - Theorem l. 10. Let A be an element of where p U such 'that is a prime of d? and ( P7 k):: I Then there exists a decomposition of the form where P l') is a primary element of rank r belonging to p, and this decomposition is unique in the sense that if also A ,:: F/'')(, where P, (,1 is a primary element of rank f' belonging to ?, then P fr) = P,''' , . shall be called the maximal left primary factor of A with respect to p . That there exists a decomposition of the form stated in the theorem is clear from Theorem 1. 2. (r1 n decompositions we have P >,,., , If P ,,.) -- P,("J P. , (rJ > A If there are two such (,.) n<t-, , and hence P "',- -, /; ? ,,., • , then p(r> .,., P'"' , is a primary element belonging to p and of rank greeter than r, a contradiction to <p~ I-<):: I. Hence the theorem. :Sy an induction we have the following theorem. Theorem 1.11. Corresponding to each order of the distinct prime factors of the norm singular matrix A l'J(A) of a non- there exists a unique - 15 - decomposition of A as the p roduct of p rima~, matrices belong ing .to the prime factors of 11.(A). Let A be an element of b having the ma.~imal Theorem 1.12 . left primary factor p ri me. psJ , ,,., of .norm p,., p a Then all primary elements which belong top and divide For if P I/ also divide P {,.) is a primary element which belong s to p and . A but does not d.ivide P~ we have divides pr,, ;, p(r;,, p/3) >4 where the first containing is proper. to This is in contradiction P~' being the maximal left :p rimary factor of 4 with respect top. A restatement of the above theorem follows. Corollary 1.12 . The maximal left p rimary factor of an el ement of [) with respect to a p rime pis the meet of the set of all its left primary factors belonging to p . ·Two elements 4 and /3 of tr will be called relatively prime if and only if their join is an element of 1, i.e. if and only if Au 8 =f. Thus two elements of O are relatively prime if and only if their only common left factors are unimodula.r elements. Theorem 1. 1 3 . Two elements 4 and B of Oare relatively - 16 - prime if and only if ri (A~ B) '=' 11/4) n(BJ. This is a consequence of Theorem 1.8 e.nd the discussion above. C'; Theorem 1.1 4 . 4 and 8 are elements of U such that If nf4) and n(f> ) a.re relatively prime elements of {~J , then )/ and .B are relative- ly prime elements of t). For 17 (,4 vB) is a factor of both r11/I-J ano_ n i/ 3) ; hence /) is a unit of //'( and ,4 ,._:(3 is u.nimodular. The converse of this theorem is not in general true. Corollary 1.14. Primary elements of 1 ferent primes of Ot Theorem 1.15. U belonging to difare relatively prime. Each element of !} is e qucl to the meet of i ts ,.maximal left primary factors. e e. i,il e,, Let il!A / -=-P, 'p. -·· ?~ 1 , where n1 is an element of [} whose di-,,) P.re.1 I?.~ maximal left primary factors are the distinct primes ?,, • • , , Di ' I " t. . , , •• , , r belonging to respectively. Since P i'e.:J > /-1/,1 , C we have By an obvious ina:uction on Corollary 1.14 and Theor em 1.13, fl But if (rj~/,,., !!,~,., -- · r1 f{e+') ~ff'J..e.... p/c ~ 11 (4 ). 8 and C are elements of tJ such that 13> C. and - 17 - Hence This representation of 4 is unique since the maximaJ. left primary factors of A are unique. Now let p, , p,. , . . . , P.· , the primes of c?. • • • be some ordering of Applying the convention mentioned just before Theorem 1.10, we mey write A wh ere -:: P. (e,J I de..; r)lt.. rl . .. ~ /!,.eJ ~ . . . Jryei:J 1s • the m<:VJ.ma ----~ 1 1 e ft pr i mary f ac t· or o f - L. rJ// with respec t 'e) to the prime ?,: . /I We will refer to /?,. as the component of ,4 with respect to A· ; only a finite number of the components of an element of () are not unimod.ular. Corresponding components of two elements of {) are components belonging to the same prime of Theorem 1.16. Let A and 8 be two elements of I) . Then A divides B if and only if the components of A divide the corresponding components of B. The sufficiency is obvious. Suppose ~A >8 , i.e. un an element of U, and let P''\e a component of .v 11 A an element of V. factor of B . Then =AC , C A , A =plrJ f( , B ~ P.-./rJ)(. 'r' p(r/ --- , and is a left primary (1') Hence by Theorem 1.12 P divides the correspond- ing maximal left primary factor of 8 . .A:n imrr.ediate corollary is the following decomposition theorem. - 18 - be decompositions of the elements 8 of 8 as the meets of their ma.·-dmaJ. ... lex t primary f a ctors (= ! , '?. , ,4 and • • • , where A te) , r.·n{e<J an a. ~ a.re co rrespond_ing compon- Thus the modular lattice f) is the direct product of the ( infinite) set of all f}A· lattices. In the case n=1 we have the well-known theorem that the lattice of elements of rf? with respect to g.c.d. and l.c.m. is the direct product of chains whose elements are powers of p rimes . of o?. It is only in this case, i.e.()-:-/, that the p rimary elements of {) are powers of primes of b only; if n7 I the p rimary elements of !) a.re not necessarily powers of primes of b, though powers of p rimes of {)- are primaries. Also it is only in the case tl=t in which 1'}(_ is commutative. We will sometimes refer to the 4.: lattices as Pn- latices, the 1 indicating the matrices of!) are n~n. The p~-lat~ces are thus the fund-a.mental lattice-theoretic structural uni ts of the modular lat t ice of non-singular l?Ul me.trices. The 11 top 11 of the - 19 - Hasse diagr am of the 3i. -lattice is shovm in Figure 1, as well a s the 2 1 -lattice 3,_ - lat t ic e . which is seen to be a sublat ti ce of the In genereJ. the p,,_ - lattice is a su.blattice of the 9"1-lattice if 1J~h? and P~9. It is conjectured that the lattice shown in Figure 2 is the lat t ice of largest order which can be generated by three elements Q, b , c. of a Ptt -lattice for n-= 2. , 1 If a'. 6 ,c. 1 are generating and p oss i bly l a r ge r values of f7 • elements of a similar lattic e , we may form the direct product of the t wo lattices. It may easily be verified. that the elements A ,:- i Q, l:) JI , 8 -- I..I D, C Ij I I I 1 J f' - '- - (C' , Q I ,)\ generate the free modular lattice of order twenty- e i ght (there a r e, of course other triples of elements of the direct product which generate the free modular lattice). The free modular lat t ice g enerated by three elements is thus a su.bdirect p roduct o:f the direct p roduct of the lattice of Figure 2 with itself. Figure 3 shows some of the elements at the top of the Hasse diagram of the23-lattice; each element of the 2~-lattice is the unit element of such a lattice. It i s seen to be a p rojective g eometl"'J, and in fact each element of a ,o~~ttice, O>I, is a unit el ement of a p rojective g eometry of rank n~,. ......:: Section II Let ct be a linear form modul of order 17 over the ring Ci? of rational integers with basis elements d, , UL , • • • , U,, . Each submodul cf £ has basis elements of the form ex} u.-1-a~Ui.+ -~ t- Cl(; ll.,i, ,· ro 1.v here the lt; are elements of V{ • If £; is a submodul of ~ having fll "'"', ... ~ such basis elements •.., f .-., '· I l\..,,'{,L{' we may write +'t..--\..1 v t.l(t.. + -· + ;,..,{ •v"u ' ..,, . or .c C\.1 -= (u, u"" I iY.., o-: LX ";... {Y ~ Thus the matrix I U,i) .\" Ct' ; ()I t. {OI / J represents the submodul £, of £ . submo d.ul of £ has a basis of n representation as an Ila?. matrix. Every elements , and h ence has a However since we will be deal- ing with ·submoduls of £ before the number of basis elements have been reduced to n., it is well to know that the norm of an matrix, and hence of the modul it represents, is the g .c.d. of the set of all 17tl2 determinants whose columns are chosen from those of the given 17 xrrt matrix (Dedekind [lJ). Since each submodul of ..p "'-- may be represented by an rt-rowed matrix, and conversely each 11.-rowed matrix ,7i th elements in <R rep resents a submodul of £ , we she.11 speak of It-rowed matrices rather than sub:nocJ:uls of £: . Thus two /l.-rowed matrices are equal if and only if they represent the same roo dul. We are interested in the case in which the basis - 23 - elements of ,:t are , , (.(J, • . . , w ~-I , where w is the root of an Ot and leadinG coef- rz.tth degree polynomial with co ef f icients in ficient unity , i. e . w is on algebraic integer. submoduls of Each pair of ,;e mey then be mu.l tiplied al gebraicly, their algebraic pro duct being the submodul of£ whose basis elements are obtained by multiplying each pair of basis elements of the two moduls. This aigehraic multiplication is described in terms of fl-rowe d matrices in the following definition. Definition 2.1. Let ,4 and /3 be rl-rowed. matrices with f' and S colmnns, respectively, and let I 'a., ) be a / x ( nH) matrix with elements in(}(, the first (a.. is called an element being unity. integral polynomial of degree fl.) Then the algebraic pro duct of A and !3 with respect to the inte gral polynomia14, which we denote by 4B , is an ,nrs matrix whose columns are obtained by multiplying each column of A with each column of B Let • , c:'.Y,· be the elements in the i I th cola..."ll!l of in the following manner. 'I 4, and 8)'., (BJ~, the elements in the j 'th column of B. • , tf - 24 - Fonn the formal polynomials where X i s an inde terminan t over (i(. The p olynomia ls lf and 18 may be formally nru.ltip lied a n d reduced modulo a.., i.e. ,,,,_, rx{:> -=:: 71< Th en / 1/ 1><., x i x J - • • -1- ~; ~ K +- 0'):: (im!)d~ , ; c, n-tn-1. -1- - 1 x I< n ~ 11< , • • • , r. '7 1K are the elements of the column of ,.:1,5 which is the :p roduct of the ;_ 1th column of A and thej • th column of ,8. The al gebraic :p ro duct of two 11-rowed. matrices A and B will not be confused ':rith row-by-column ma tric multiplication discussed. in Section I, which, in this section, will be indicated by 4 -rB. A and 8 To illustrate al gebraic mu.l tiplication of matrices let be quadratic moduls 4 , . ( '2., l+W) > B =- ( 3~ 2+.:zc.u) {in Dedekind's notation) where~ is a root of 1, a., -.: u.> -t- W+ J. Then - 25 - This mcy be reduced to canonical form to obtain In matric notation , ' 8 -_!3Z)\ l O 2. In nru.l tiplying the columns of second column of 4 and .B only the product of the .4 with the second column of B need be reduced_ modulo 4. Thus X+--/ ( \:'. '=' f3 :- Z(-1- 1 ) ll-::X+x+/ ,) Hence AB ::, 48 -:, ( e:, 4 3 ~ ( , 4 3 I '2. I \. C I ). 0 \ '2._ / , - 26 - i'te recorcl the following well-k nown p roperties of al gebraic multiplication all of which go back to Dedekind. . /l , ,B , C are any /{-rowef:1 matrices. .4 B (i) (Closure) is an~ -rowed matrix. (±1.) (Associativity) (iii.) ( Connrru. ta ti vi ty) ( iv) ,// ::: B _ _.,.... 4 C' .., 13 C . (v) /-1(8 uC) -:;; ,4B t_·,4C. ( JI- B ) (!, =-JI ( BC). I/ 8--= /34. (iv) and (v) mey be use d. to deduce ,4 .>8->-·/-/C > BC . fl -rowed matrix / J will be called a ring if matrix ,4 ~ A ,-,4 . An An -rowed will be called_ an ideal if (vi) I 4 =4 , where I is the unit,1-rowed matrix. The real tion between multip licative and. lattice-tl~eoretic p rop erties of ideals is strong due mainly to (vi). A study of the abstract p roperties of an arbitra r y lattice over which a multiplic a tion satisfying ( i) through t (vi) may be defined has been initia t ed by Ward. a nd furthered by Ward and Dilworth 11]. If A is an ideal, t hen A :> 4/3 , ,end this p roperty may be used to intro duce a residuation in the lattice of i o.eals . Howeve r if /J ancl B are /[-rowed matrices, not idea ls, it is not in general true th.at A ;> 4B . may occur: A) ,48 related to 413 In fact any one of the following p ossibilities , or 4:::/1-B , or AB ?// It is not difficult to find through containing . algebraic mo dul s which sa tisfy ,4-:;; A , or JI mey not be ~ , s~I L}p , while ,4 ;J- , • if - 27 - J <rd; in fact this p rop erty of forming a cycli c g r ou_p of finite orcler und er al gebraic multiplica tion p robably hold s for all n-rowed matrices s a tisfying I4':# I , i.e. r'l.-ro,ved matrices which 11 gene rate 11 the U11it ma trix. is any ;z-rowed matrix. For FA is a lways an idea l, whe r e / JI T/J ..1..-t-lI c HJ - j- L4 , "" -,-'"'T ;' /-} // We s ay tha t T4 is the id_eaJ. that A generates. is an i d. eal if and only if it gene rates it s elf. In this connection the follo wing interestinc but as ye t unan swered question may be r a ised. Suppose A is an i deal such that h= BC where B crid C a re non-unit ideal s , a nd su.-ppos e that i '.' • e . I" /--I, i. /I,:::- /.I • Cr such that /4-::: ,/4 is an n-ro wed matrix which gene rates Then do there exist 11-rowea_ matrices B, and B 1 A few of the C, v;here E3, generates 8 and C, g enerates C ? peculiarities of the multiplica tive p rop erties of general,1-rovred matrices a re given above to emphasize the weakness of the relationship b e tween their multiplicativ e and lattice theoretic prop erties. It is therefore all the more surprising that there is a certain r ogi__i_lari t:t in thi s resp ect based. upon the d ecomposition Theorem 1.17. We wHl need the following lemmas Lemma 2.1. Let ,4 be on 11-rowed matrix such that all the elements of a column of A have a factor, say k, in connnon, and let ,4, be the 17-rowed matrix which 4 becomes on deleting the factor ,/A) I kn l4,J. . Then - 29 - 1 1 where the multi-plication (e. g . t ' Q, ) is the usual scalar multiplication of matrices by elements of u?, and_ the ad.di tion is term f or term addition of matrices. Then 48 -=-ca,6,,a,&, ... ,a,6,,;at.6,J •.. ,a,.b,,, • •• ;a.,"D,,a.,,b.., • ·· ;a,,Li.) C.D = (c.,d., eda ~.: .. C, d,, c..,_d, ,c;cl,, ,C!n d,. c,,d . . , ••• )c,,,d,,), 5 J - • • J j J ••• where the nml tiplications a~~ ·end d:d1 are the polynomial multiplications described in Df. 2.1 reduced modulo some fixed inte gral polynomial. The reI:Iainder of the p roof consists in repeated application of Lemma 2. 1. Thus, since we h ave I ~,,1C'/'' n(a,b,,e,d, .. · .. ,c,d.,,c..'l.d .•• •· ;c11d"). n(c_DJ But • 1,1 trf11.. , Hence, subtracting (: y,- , , times the first column from the second column, and then applying Lemma 2.1, But e,d 3 ~ e C: 3 a., b, + f ,14>,a~.6 ..+ f 1 1 ' 1 1 ' . Hence i f we subtract the 1 i ne ::i.r combination 11 ' cp a, b', ·h::,.,/(,t,3a,01 1 L,,11,3 c 3 <P?,' a,b5. 1 ..,.. of the f irst two columns f rom t he third column, end then apply Lermna 2.1, we obtain n (tD) l (e1' 1)\p111 cif'1ev3~ 0 (o.,b.,a,h./t,~.C.c.\-1-, · •.,Gd., .,l!,_d, .... ,ct\d'l). - 30 - '!'he p rocedure should. now be clear . For exampl e after the f irst n co lumns hav e been thus tr ea ted we find Continu ing in th i s fash ion we finally ob t a in h (CD) I l~·~'l (61':J'1-··lt:"·~/'(cp '•'epi,i_.. (f·, 0 n (,4 B) . as was to be shoYm . Theo r em 2 . 1. The algeb r a i c p roduct of t wo pr i mary matri ce s belonging to the same p rime pis also a pri mary matrix belonging top . . . P1 0 , 2~ be p rimary matrices belonging to the p ri me p , Let .i and. as sume they a re in canonical form, i. e . p,.· CY i,i. Pr'" ' .,, l (Y1., '1 P(i'J ::: I PS, ,3 1,1.. (X ' p3z. ') - P.1.(3)=,,, p p r,, wh ere some I',; and. ~- may be zero , bu t /J 2 r - - I' , 4 - Now in ,, the algebra i c p ro duct the follo wi ng fl columns occur p"'• {31,'l.. pr:;s •·'!. - . pr. .; ~, p ".+ ::.1,. p l';/,'1 p"'(3°2.,3 Pr-'(3~,., pli+::S. 3 P "'l33," p 'i t-S,, where no reduction modulo the f ixed polynomial is nec essa ry . But th e det erminant having thes e ,,..."ifs 1• . e . ,~ , an d n \1. P.("' , /;..")is J; ' 2. /7 columns is a power of p , a.ivides this determinant . Hence - 31 - II ( P,"1P,(s;) is a p owe r of p , erco the theorem. The follo n i ng theorem s t a t e s a r emarkabl e nnil tip lic a t i ve prop erty of t h e decomp os ition of Theo rem 1.15 in adci.i tion to the laftice-theo retic result of Theo r em 1.17. The oren .2 . 2. A Let and B be non- s i nt,u l a r 11 - rowed matrices having the d eco ln_9ositions A - [) ~,),.-, _Dif-'t,) ,-, ... .-, ,o,_f!r) /l - • - I J • I 3, l t:;e described in Theor em 1.15. Then 48 =1~tJQf'J,, 1Y/•1{J._t..J-4 .. .., R.~c.:fA-J, ... . i . e . the componen ts of the alg eb r a ic p ro duc t of 4 and 8 a r e the a l g ebra i c p ro duc ts of corresponcUng components of 4 and 8. Since (eJ P.f I! ) I-#, QfJ > B, i t f ollo ws n{e.-J .,(fJ Thus It ' 4-i, wh i ch i s a p rimary ~a tri x belonging to the p rime b y Theorem 2.1 is a left f acto r of AB. is th e maxi mal left p rimary f a ctor of /+B with r espect to we n ee d only show that p owe r of r0 : th an n (AB ) A·1e,:J,,-:, '--' ..m.-JJ' 1 f7l 1 To p rov e that ffedc,C ,~ p ,· i s not divi s i b le by a l arger Let 4, "'A , 23, =-B , where - 32 - A, .snd 8, are canonical matrices. Then Therefo re by Lemma 2. 2 r n C4B) I 11 tJ n(D) J1 fl cr?d Qft:)) , where p; is not a factor of [n(C)11(D)] th en 11 • Hence if p/l n (4B) rp/' I n CR~dQfftdJ , and the theorem follows . .An immediate corollary of Theo rer:i 2. 2 is the following theorem. Theorem 2. 3. A matrix is a ri ng or ideal if and only if each of its components is a ring or ideal, respectively. Theorem 2.2. and the above theorem indicate tha t in making a further study of the multiplicative p roperties of al gebra ic moduls, rings , and i deals it is sufficient to conf ine attention to the ?n-lattices. Thus the .Rt-lattices a re the fundamental units of structure of algebraic moduls for multiplicative as \'fell as lattice-theoretic purposes. A curious corollary of Theorem 2.2 . is t he followin g . Theorem 2.4. If t wo relatively p rime matrices A and 8 are such that ( '7 (A) , 11 {8) ,48 ) -=- / , then is an ideal. For, since ( n{,4) , II (8) )" /, a t least one component of - 33 - each pair of correspond. ing component s of Hence each compon ent of the p:i:od.uct 48 is also. Notice tha t algebraic number ring. AB A and B is unimodula r. is a n i deal , a nd hence - A and B are in any n' th decree Thus, for exainple, the p roduct o:f primary matrices belonging to distinct p rimes is an idecl, alweys. References G.Birkhoff 1. Lattice Theory, 1940. A. Ch-atelet 1. Groupes abeliens ~ Paris, 1924. R. Dedekind 1. Gesammelte mathematische ~ ' Dritter Band., Ober die Theorie de r ganzer aJ.gebraischen Zahlen. 2. Ges ammel te mathematische ~ . Zwei ter Band, fl°ber die von drei Moduln erz eu gte Dualg rupp e. c. c. MacDuffee 1. The Theory .£! Matrices (Ergebnisse der Mathematik und ihrer Grenzgebiete), Springer, 1933. B. L. Van der Waerden 1. Moderne Al gebra , Zwei ter Teil, (Zweite Auflage), Chap ter XV. Also Chap ter XIV on Ganze aJ. gebra ische G~ssen. M. Ward and R. P .Dilworth 1. Risidua ted Lattices, Tran s . luner. Math. Soc. vol. 45 (1939) pp . 335-354.