Greatest Common Divisor and Least Common Multiple Matrices on Factor Closed Sets in a Principal Ideal Domain

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1 Joural of Mathematcs ad tatstcs 5 (4): , 009 IN cece Publcatos Greatest Commo Dvsor ad Least Commo Multple Matrces o Factor Closed ets a Prcpal Ideal Doma AN El-Kassar, Habre ad 3 YA Awad Lebaese Amerca Uversty, PO Box , Choura-Berut, 0 80, Lebao Lebaese Amerca Uversty, Lebao 3 Lebaese Iteratoal Uversty, Lebao Abstract: Problem statemet: Let T be a set of dstct postve tegers, x, x,, x The matrx [T] havg (x, x ), the greatest commo dvsor of x ad x, as ts (,)-etry s called the greatest commo dvsor (GCD) matrx o T The matrx [[T]] whose (,)-etry s [x, x ], the least commo multple of x ad x, s called the least commo multple (LCM) matrx o T May aspects of arthmetcs the doma of atural tegers ca be carred out to Prcpal Ideal Domas (PID) I ths study, we exted may recet results cocerg GCD ad LCM matrces defed o Factor Closed (FC) sets to a arbtrary PID such as the doma of Gaussa tegers ad the rg of polyomals over a fte feld Approach: I order to exted the varous results, we modfed the uderlyg computatoal procedures ad umber theoretc fuctos to the arbtrary PIDs Propertes of the modfed fuctos ad procedures were gve the ew settgs Results: Modfcatos were used to exted the maor results cocerg GCD ad LCM matrces defed o FC sets PIDs Examples the domas of Gaussa tegers ad the rg of polyomals over a fte feld were gve to llustrate the ew results Cocluso: The exteso of the GCD ad LCM matrces to PIDs provded a lager class for such matrces May of the ope problems ca be vestgated the ew settgs Key words: GCD matrx, lcm matrx, factor-closed sets, prcpal deal doma INTRODUCTION Let T = {x, x,, x } be a set of dstct postve tegers The matrx [T] havg (x, x ), the greatest commo dvsor of x ad x, as ts (,)-etry s called the Greatest Commo Dvsor (GCD) matrx o T The matrx [[T]] whose (,)-etry s [x,x ], the least commo multple of x ad x, s called the least commo multple (LCM) matrx o T The set T s sad to be factor closed (FC) f t cotas every dvsor of x for ay x T I 876, mth [] showed that the determat of the GCD matrx [T] o a FC set T s the product = ϕ(x ), where ϕ s Euler's totet phfucto Moreover, mth cosdered the determat of the LCM matrx o a FC set ad showed that t s equal to the product = ϕ(x ) π(x ), where π s a multplcatve fucto defed for a prme power p r by π(p r ) = p ce the may papers related to mth's results have bee publshed Recetly, ths feld has bee studed tesvely Ths ew sprato started by Besl ad Lgh [3,4] I [3], Besl ad Lgh obtaed a structure theorem for GCD matrces ad showed that, f s FC, the Correspodg Author: AN El-Kassar, Lebaese Amerca Uversty, PO Box , Choura-Berut, 0 80, Lebao Tel: , Ext 89 Fax: det[t] = ϕ(x ) They coectured that the coverse s = true I [0], L proved the coverse ad provded a formula for the determat of a arbtrary GCD matrx Besl ad Lgh [4,5] geeralzed these results by extedg the FC sets to a larger class of sets, gcdclosed sets I [], a structure theorem for [[T]] was obtaed from the structure of the recprocal GCD matrx /[T], the (,)-etry of whch s /(x, x ) Gve a FC set T, Bourque ad Lgh [6] calculated the verses of [T] ad [[T]] ad showed that [[T]][T] s a tegral matrx I that study, they stated ther famous coecture that the LCM matrx o ay gcd-closed set s vertble Bourque ad Lgh [7,8] vestgated the structures, the determats ad the verses assocated wth classes of arthmetcal fuctos For a bref

2 J Math & tat, 5 (4): , 009 revew of papers relatg to mth's determat, we refer to [9] Usg the laguage of posets, the authors gave a commo structure that s preset may extesos of mth s determats Besl ad El- Kassar [] exteded the results [3] to uque factorzato domas The purpose of ths study s to exted may of the recet results cocerg GCD ad LCM matrces defed o factor-closed sets to arbtrary Prcpal Ideal Domas (PID) such as the doma of Gaussa tegers ad the rg of polyomals over a fte feld MATERIAL AND METHOD Let be a PID ad let a, b We say that a ad b are assocates ad wrte a~b, f a = ub for some ut elemet u If b s a ozero out elemet, the b has a uque factorzato, up to assocates, to prme α α α elemets That s, b = up p p, where the p s are dstct prmes Also, every fte set {b, b,, b } admts, up to assocates, a greatest commo dvsor For a ozero elemet b, defe q(b) to be /<b>, the order of the quotet rg /<b>, where <b> s the prcpal deal geerated by b Note that q(u) =, for ay ut u Also ote that Z, Z[] ad Z p [x], q(b) s fte b 0 Throughout the followg we cosder to be a PID havg the property that q(b) s fte b 0 It ca be show that q(ab) = q(a)q(b) Hece, f α α3 α α α α b = up p p, the q(b) = (q(p )) (q(p )) (q(p )) For a postve teger, q() = ad for a Gaussa teger a+b, q(a+b) = a²+b² Also, f f(x) s a polyomal of degree Z p [x], the q(f(x)) = pⁿ Defe φ s (b) = U(/<b>), the order of the group of uts U(/<b>) The φ s (b) ad the equalty holds ff b s a ut Also, φ s (b) = q(p) ff p s prme It ca be show that φ s (b) s multplcatve ad α 3 f b up α α = p p, the: ϕ (b) = (q(p ) )(q(p )) (q(p ) ) α (q(p )) (q(p ) )(q(p )) α α () Now, f = Z, the φ s (b) becomes Euler's phfucto Also, f f (x) = (p α α α (x)) (p (x)) (p (x)) s a polyomal of degree = Z p [x], a product of powers of dstct rreducble polyomals p (x),, the: ϕ r k k (αk -) rk s(f (x))=p = (p ) k= () where, p (x) s of degree r For example, f f(x) = (x²+ ) 4 (x+) 4 (x 3 +x +x+4) = Z [x], the φ s (f(x)) = Now, f β = u β k k k β β, a product of dstct Gaussa prmes β = a +b,, the: k ( ) ( ) φ ( β)= a + b a + b s = For example, f β = 6+4 ~ 3(+) 3 (+), the φ s (β) = 640 Let be a PID ad let T = {t, t,, t } be a set of ozero oassocate elemets Defe a lear orderg o T accordg to the followg scheme: If q(t ) < q(t ) the t t ad f the equalty q(t ) = q(t ) holds the order t ad t accordg to ay scheme depedg o the gve doma For stace, f = Z[] ad q(t ) = q(t ), where t ~ a+b, t ~ c+d, a, b, c, d 0, the defe t t wheever b < d I the case = Z p [] ad q(t (x)) = q(t (x)), where t (x) ~xⁿ+a xⁿ ++a x+a 0, t (x) ~ xⁿ+b xⁿ ++b x+b 0, 0 a, b p-, 0 s the smallest dex such that a b, the defe t (x) t (x) wheever a 0 <b 0 If the set T s ordered so that t t t, we say that T s q-ordered Two sets T ad T are assocates, deoted by T~ T, ff each elemet T s assocate to a elemet T ad vce versa For a ozero elemet b, let E(b) be a complete set of dstct oassocate dvsors of b The, E(a) E(b) ~E((a, b)) ad E(p m ) ~{, p, p,, p m } Note that f t~t, the q(t) = q(t ) ad φ s (t) = φ s (t ) Also, φ (d) = φ(d) wheever T~ T d T d T ' Theorem : Let be a PID ad let b be a ozero elemet If E(b) s a complete set of dstct oassocate dvsors of b, the q(b) = φ(d) d E(b) Proof: Let b 0 The result s true whe b s a ut α α α uppose that b s a out so that b = up p p ce φ s s multplcatve, the fucto f (b) = ϕ (d) d E(b) s also multplcatve For ay prme elemet p, () gves: f (p ) = ϕ (d) = ϕ (p ) + ϕ (p ) + + ϕ (p ) 0 d E(p ) = + (q(p ) ) + (q(p ) )(q(p )) + + (q(p ) ) (q(p )) (q(p )) q(p ) = = 343

3 J Math & tat, 5 (4): , 009 By the multplcatvty of f(b), we have q(b) = ϕ(d) d E(b) Corollary : (Euler's) If s a postve teger, the = ϕ(d) d> 0,d REULT GCD matrces o FC sets a PID: Throughout the followg, we cosder T = {t, t,, t } to be a q- ordered set of ozero oassocate elemets of a PID Defe the GCD matrx o to be the matrx [T] = (t ) = q((t, t )) The set T s sad to be a factorclosed (FC) ff t T ad d t mples that d~ t for some t T Note that ay set T s ether factorclosed or t s cotaed a factor-closed set D Theorem : The GCD matrx [T] ca be decomposed to a product of a m matrx A ad a m matrx B, for some m The ozero etres of A are equal to φ s (d) for some d a FC set D cotag T ad B s a cdece matrx Proof: Let D = {d, d,, d m } be a FC set cotag T the PID Defe the m matrx A = (a ) by ϕs(d ) f d E(t ) a = ad let B = (b ) be the cdece 0 otherwse matrx correspodg to the traspose of A, where f a 0 b = Hece, the product AB s gve by: 0 f a = 0 dk E((t,t )) (AB) = a b = ϕ (d ) = ϕ (d ) = k k s k s k k = dk E(t ) dk E(t ) E(t ) dk E(t ) ϕ (d ) = q((t,t )) s k Example : Let T = {,+x,+x 3,(+x) 3 } Z [x] The [T] = Let D = {, +x, (+x), 8 8 +x+x, +x 3, (+x) 3 } The, [T] 4 4 = A 4 6 B 6 4 where: A = ad B = Example : Let T = {,, 5} I Z[], [T] = 4 5 Note that T s ot FC Z[] elect D = {, +,, +, , 5} The A = ad B = Theorem 3: The GCD matrx [T] s the product of a m matrx A ad ts traspose A T The ozero etres of A are of the form ϕ (d) for some d a FC set D cotag T Proof: Let D = {d, d,, d m } be a FC set cotag T Defe the m matrx A = (a ) by ϕs(d ) f d E(t ) a = Hece, the product A A T s 0 otherwse gve by: T k k ϕs k ϕs k k = dk E(t ) dk E(t ) (AA ) = a b = (d ) (d ) = ϕ (d ) = ϕ (d ) = q((t,t )) s k s k dk E(t ) E(t ) dk E((t,t )) Note that Theorem ad 3 hold eve f T s ot q- ordered I the case whe both T ad D are q-ordered, B becomes raw-echelo form Corollary : (mth's Determat over a PID) If T s FC, the [ ] det T = ϕ (t ) = Proof: Let T be a FC set Choose D to be q-ordered ad D ~ T From Theorem, the GCD matrx [T] = AB, where A s a lower tragular matrx ad B s a upper tragular matrx such that a = φ s (t ) ad b =, Therefore, det[t] = det[ab] = det[a]det[b] = ϕ(t ) ϕ(t ) ϕ (t ) We ote that f T = {t, t,,t } s ay arragemet of the elemets of T = {t, t,, t }, the det[t] = det[t ] Ths ca be verfed as follows 344

4 J Math & tat, 5 (4): , 009 The matrx [T] ca be obtaed from [T ] by swtchg the rows ad the colums of [T ] Thus, [T] = E E E [T ], where the E s are elemetary matrces wth det[e ] = ±, Hece, [T] ad [T ] are smlar matrces ad det[t] = det[t ] Next, we cosder the coverse of Corollary Let be a PID ad let T = {t, t,, t } be a oempty set of ozero oassocate elemets wth [ ] Is t true that T s factor-closed? = det T = ϕ (t ) Cosder a mmal FC set D = {t, t,, t, t +,, t +r } cotag T = { t, t,, t } wth t t t ad t + t + t +r Defe a (+r) matrx A by (A) = ε ϕ (t ), where ε s f t E(t ) ad 0 otherwse Deote the matrx (ε ) (+r) by E, a {0,}- matrx Note that the matrx A s the same matrx A defed Theorem 3 For a m matrx M, > m ad ay set of dces k, k,, k wth k < k < < k m, let M deote the submatrx cosstg of k th, k th, (k,k,,k ) k th colums of M Theorem 4: Let D = {t, t,,t, t +,, t +s } be a mmal FC set cotag T = { t, t,,t }, where t t t ad t + t + t +s The: k < k < k + s ( ) det[t] = det[e ] ϕ (t ) ϕ (t ) ϕ (t ) (k,k,,k ) s k s k s k Proof: ce [T] = AA T, Cauchy-Bet formula gves that: ] [ = T det T = det AA = k < k < k + s k < k < k + s T ( det[a (k ),k,,k )]det[a(k,k,,k )] ( det[a (k ),k,,k )] The result follows from the fact that: det[a = det[e ] ϕ (t ) ϕ (t ) ϕ (t ) (k,k,,k ) (k,k,,k ) s k s k s k Corollary 3: Let [T] be the GCD matrx defed o T The, det[t] ϕs(t ) ϕs(t ) ϕ s(t ) Proof: The terms the summato of Theorem 4 are oegatve ce the submatrx E (k,k,,k ) s lower tragular wth dagoal elemets equal to, we have that the term correspodg to (k, k,, k ) = (,,, ) s det[e(,,, )] φ s (t ) φ s (t ) φ s (t ) = φ s (t ) φ s (t ) φ s (t ) Therefore, det[t] ϕs(t ) ϕs(t ) ϕ s(t ) Theorem 5: Let [T] be the GCD matrx defed o T The, det[t] = ϕs(t ) ϕs(t ) ϕs(t ) f ad oly f T s factor-closed Proof: The suffcet codto holds from Corollary Coversely, suppose that det[t] = ϕs(t ) ϕs(t ) ϕ s(t ) For cotradcto purposes, suppose that T s ot FC Let D = {t, t,, t, t +,, t +s } be a mmal FC set cotag T such that t t t ad t + t + t +s ce T s ot FC, D s ot assocate to T The, t + s D but ot T ad t + E(t) for some t T Now, let t r be the frst elemet T such that t + E(t r ) The, the submatrx A(,,r, +,r +,,) cosstg of the st, d,, (r ) th, (+) th, (r+) th, ad th colums of A (+s) s a lower tragular matrx of ozero determat Hece, E + + s a {0, }-matrx whose dagoal (,,r,,r,,) elemets are equal to ce E(,,r,r +,,, + ) ca be obtaed from E(,,r, +,r +,,) by performg a certa umbers of successve colum permutatos, det[e (,,r,r +,,, + ) ] = ± det[e (,,r, +,r +,,) ] = ± From Theorem 4, we have: k < k < k + s ( ) det[t] = det[e ] ϕ (t ) ϕ (t ) ϕ (t ) (k,k,,k ) s k s k s k = ϕ (t ) ϕ (t ) ϕ (t ) + ϕ (t ) ϕ (t ) ϕ (t ) ϕ (t ) s s s s s s r s r + ϕ (t ) ϕ (t ) + > ϕ (t ) ϕ (t ) ϕ (t ) s s + s s s Ths cotradcts the ecessary codto that the equalty holds Iverses of GCD matrces a PID: Let t be ay ozero elemet The geeralzed Mobus fucto over s defed by: µ = f t sa ut m s(t) (-) f ts the product of m oassocate prmes Note that: µ s(d) = d E(t) 0 otherwse f tsa ut (3) 0 otherwse Corollary 4: Let [T] be the GCD matrx defed o T The, [T] s vertble ad ts verse [T] = (r ) s gve by: 345

5 J Math & tat, 5 (4): , 009 r = µ (t / t ) µ (t / t ) s k s k t k ) ϕs(t k) t E(t k ) Proof: Defe the matrces E = (e ) ad f t E(t ) U = (u ) as follows: e = ad 0 otherwse u µ (t / t ) f t E(t ) 0 otherwse s = The, (EU) f t ~t = e u = k k k = s(t / t ) s(t k ) The last t E(t k ) tk E(t / t ) 0 otherwse µ = µ = equalty follows from (3) ce the elemets T are oassocate, we have U = E If D s the dagoal matrx dag(φ s (t ), φ s (t ),, φ s (t )) ad A = ED /, the [T] = AA T = (ED / )(ED / ) T = EDE T Therefore, [T] = U T D U = (r ), where r = (U T D U) = u u = µ (t / t ) µ (t / t ) k k s k s k k= ϕs(t k) t k ) ϕs(t k) t E(t k ) Example 3: Let = Z [x] ad let T = {,+x,+x, (+x) 3,+x+x }, whch s a q-ordered FC set of ozero oassocate elemets Z [x] The, [T] = 4 4 By corollary 4, [T] s obtaed as follows: 7 a = + + = + + = () ( x) ( x x ) 3 3 ϕ ϕ + ϕ + + a = = ϕ ( + x) a = a = 0, a = = ( x x ) ϕ + + ad so forth Therefore: of T yeld smlar recprocal GCD matrces For a ozero elemet t T, defe the fucto ξ by ξ (t) = q(d) s(d) q(t) d µ A geeralzed verso of the E(t) Mobus verso formula ca be used to show that = ξ(d) ce ξ(t) s the product of two q(t) d E(t) multplcatve fuctos ad χ (t) = q(d) µ s(d), q(t) d E(t) we have that ξ(t) s tself multplcatve Moreover, f p s prme, the χ(p ) = q(p) Hece, q(p) ξ (p ) = Therefore: (q(p)) ϕ (t) ξ = = (4) (t) ( q(p)) ( q(p)) q(t) p E(t) (q(t)) p E(t) where, the product rus over all prme dvsors p of t E(t) I the followg two theorems we obta two factorzatos for the recprocal GCD matrces Theorem 6: Let D = {d, d,, d m } be a FC set cotag T The recprocal GCD matrx defed o T s the product of a m matrx A = (a ), defed ξ(d ) f d E(t ) by a = ad a m cdece matrx 0 otherwse B correspodg to A T Proof: Let A be as defed ad let B be the m matrx f a 0 wth b = The: 0 f a = 0 m (AB) = a b = ξ (d ) = ξ(d ) k k k = d E(t ) d E(t ) E(t ) d E(t ) = ξ (d ) = q((t,t )) d E((t,t )) 7 / / 3 3 / / / 4 / 4 0 / / 3 [T] = 0 / 3 / 4 / 4 0 Recprocal GCD Matrces a PID: The recprocal GCD matrx o T s the matrx /[T] whose (,)-etry s /q((t, t )) It s clear that /[T] s symmetrc Furthermore, permutatos of the elemets 346 I a smlar maer we prove the secod factorzato gve the followg theorem Theorem 7: Let D = {d, d,, d m } be a FC set cotag T ad let C be the the matrx gve ξ(d ) f d E(t ) by a = The /[T] = CC T 0 otherwse The proof of the followg theorem s smlar to those of Theorems 4 ad 5

6 J Math & tat, 5 (4): , 009 Theorem 8: Let T be a set The, det(/[t]) = ξ(t )ξ(t )ξ(t ) ff T s factor-closed polyomals over a fte feld may gve ew sght to some ope problems LCM Matrces o FC ets a PID: The least commo multple (LCM) matrx defed o T s the matrx [[T]] = (t ), where t = q([t, t ]) ad [t, t ] s the least commo multple of t ad t Theorem 9: If T s FC, the det [ T ] = ϕ(t ) ( q(p)) = p E(t ) Proof: ce [t, t ]~(t t )/( t, t ), we have ad q([t, t ]) = q(t )q(t )/q(( t, t )) Now q(t ) ca be factored out from the th row ad q(t ) from the th colum to obta /[T] Hece, [[T]] = D(/[T])D, where D s the dagoal matrx wth dagoal etres q(t ), q(t ),, q(t ) From 4, we have that: det[[]] = det(d(/[t])d) = det(d)²det(/[t]) = (q(t ))²(q(t ))²(q(t ))²ξ(t )ξ(t )ξ(t ) = ϕ (t ) ( q(p)) = p E(t ) Cauchy Bet formula yelds a formula for the determat of the LCM matrx defed o a set T whch s ot ecessarly FC The formula s gve by: det[[t]] = k < k < k + s ( ) ( (k ),k,,k ) det E q(t )q(t )q(t ) ξ(t ) ξ(t ) ξ(t ) k k k k k k From Theorem 9, we have that the determat of the GCD matrx o dvdes the determat of the LCM matrx wheever T s FC DICUION Most of the exstg results related to GCD ad LCM matrces are obtaed the doma of atural tegers The results are based o certa umber theoretc fuctos such as Euler s ph fucto ad the Mobus fucto These fucto ad ther propertes ca be geeralzed to prcpal deal domas By descrbg the uderlyg computatoal procedures ad the varous propertes the ew settgs, the exstg results related to GCD ad LCM defed o factor closed sets are exteded to PIDs Ths provdes a large class of such matrces where may ew examples ca be costructed I partcular, examples the domas of Gaussa tegers ad the rg of 347 CONCLUION The exteso of the GCD ad LCM matrces to PIDs provde a lager class for such matrces May of the ope problems ca be vestgated the ew settgs For future study, we suggest the problem of extedg GCD ad LCM matrces defed o gcd closed sets to PID REFRENCE Besl,, 99 Recprocal GCD matrces ad LCM matrces Fboacc Q, 9: Besl, ad N el-kassar, 989 GCD matrces ad mth's determat for a UFD Bull Number Theor Relat Top, 3: 7- html?pg=ii&s=645 3 Besl, ad Lgh 989 Greatest commo dvsor matrces Lear Algebra Appled, 8: DOI: 006/ (89) Besl, ad Lgh, 989 Aother geeralzato of mth's determat Bull Aust Math oc, 40: Besl, ad Lgh, 99 GCD closed sets ad the determats of GCD matrces Fboacc Q, 30: Borque, K ad Lgh, 99 O GCD ad LCM matrces Lear Algebra Appled, 74: DOI: 006/ (9) Borque, K ad Lgh, 993 Matrces assocated wth classes of arthmetcal fuctos J Number Theor, 45: DOI: 0006/th Borque, K ad Lgh, 993 Matrces assocated wth arthmetcal fuctos Lear Multlear Algebra, 34: 6-67 DOI: 0080/ Haukkae, P, J Wag ad J llapaa, 997 O mth's determat Lear Algebra Appled, 58: 5-69 DOI: 006/ (96) L, Z, 990 The determats of GCD matrces Lear Algebra Appled, 34: DOI: 006/ (90)900- mth, HJ, 876 O the value of a certa arthmetcal determat Proc Lod Math oc, 7: 08- DOI: 0/plms/s-708