THE CHAIN CONDITION OF MODULE MATRIX

Transcription

1 Jural Karya Asli Loreka Ahli atematik Vol 9 No (206) Page Jural Karya Asli Loreka Ahli atematik THE CHAIN CONDITION OF ODULE ATRIX Achmad Abdurrazzaq Ismail bi ohd 2 ad Ahmad Kadri bi Juoh Uiversiti alaysia Perlis Istitute of Egieerig athematics Kampus Pauh Putra Arau Perlis alaysia 2 Associate ember Istitute of athematical Research UP Serdag Selagor alaysia Abstract : odule is a algebraic structure built from a Abelia group with scalar multiplicatio of rig We say that modules is Noetheria modules if every submodules of satisfy the ascedig chai coditio Let R R be a rig ad ( ) is set of matrix which the etries are elemet of R The aim of this study is to perform algebraic structure from set of matrix that satisfied modules structure Furthermore we study about coditio for set of matrix that satisfies the Noetheria module ad also some result of this Noetheria coditio for module matrix ( R) Keywords: odule Noetheria Noetheria odule odule atrix 0 Itroductio I this paper we study about algebraic structure built from set of matrix which is module structure A matrix is rectagular array of umber or expressio arraged i rows ad colums (Ato & Rorres 2005) atrix is part of liear algebra which ofte used to solve some liear equatio problems I this study we used matrix with the etries elemet of commutative rig the etries of the matrix is the essetial part i this study Algebraic structure is set together with biary operatios that satisfy some axioms A operatios is said to be biary o a set A if for every two elemets of A the result of the operatio is also elemet of A The type of the algebraic structure depeds o the axioms of the biary operatio is fulfilled that is commutative associative distributive idetity elemet ad iverse elemet (Fraleigh 2003) Geerally algebraic structures cosist of group rig ad module Let G be a oempty set ad S is subset of G G together with a biary operatio is said to be group if the biary operatio satisfy associative distributive idetity elemet ad iverse elemet for every elemet of G Furthermore if the commutative axiom also fulfilled o set G the it is called Abelia group But if the biary operatio oly satisfies associative axiom the the algebraic structure called semigroup Notice that for subset S together with biary operatio of G it is called subgroup if its elemet satisfies every group 206 Jural Karya Asli Loreka Ahli atematik Published by Pustaka Ama Press Sd Bhd

2 Jural KALA Vol 9 No Page axiom Ad for subrig ad submodule have a similar cocept to subgroup (Adki & Weitraub 992) Let R be a oempty set together with two biary operatios suppose that additio ad multiplicatio is said to be algebraic structure rig if R together with additio is a Abelia group ad R together with multiplicatio is a semigroup Furthermore if R together with multiplicatio satisfies the Abelia group the R called field oreover a Abelia group is called module whe we defied scalar multiplicatio betwee ad rig R such that the scalar multiplicatio satisfies a ( m + ) = am + a ( a + b) m = am + bm ( ab) m a ( bm) ad R m (Camero 2008) = = m for every a b elemet of rig ad m elemet of Abelia group Defiitio (Grillet 2007): The subsets Si S ( i = 23 K ) are said ascedig chai coditio (resp descedig chai coditio) if the chai of subsets S S2 S3 becomes statioary i the sese that there exist m = 23 K such that S = S m Let be module over rig R ad also i m ( i = 23 K ) submodules of odule called Noetheria module if for every submodules of satisfy the ascedig chai coditio (Defiitio ) Study about module theory ad matrix theory has bee doe util recet days ad also study about algebraic structure of matrix rig set of matrix as the set together with additio ad multiplicatio matrix still beig the mai iterest by may researcher util recetly However study about algebraic structure modules with matrix as the mai set is difficult to fid ot eve foud o some iteratioal jourals oreover study about modules theory especially Noetheria modules also rarely bee foud i the last 0 years Naghipour (2005) used Noetheria module to support his study about Ratliff-Rush closures Afterward i 202 ad 205 study about Noetheria modules had bee doe agai about some properties related Noetheria modules (Aldosary & Alfadli 202; Cuog Quy & Truog 205) This shows that study about Noetheria modules becomes less iterested i Besides that study about algebraic structure matrix rig has bee cotiually assessed but study about modules matrix are ot foud This paper studies about algebraic structure module with set of matrix as the mai set Furthermore we ivestigate more about the coditios of the set of matrix so that ca satisfy the ascedig chai coditio ad the result or cosequece of this coditio for set of matrix 2

3 Achmad Abdurrazzaq et al 20 Theoretical Backgroud This sectio will preset some theories related to algebraic structures to be used for solvig problems i this study Defiitio 2 (Adki & Weitraub 992): Let G = ( G ) for G G is called group if its closed uder biary operatio ad the followig axioms are satisfied (i) For a b c G we have ( a b) c = a ( b c) (ii) There exists e G such that for all x G e x = x e = x (iii) For a G there is a uique iverse elemet a ' G such that a a ' = a ' a = e Group G is called Abelia group if it satisfied the commutative axiom Defiitio 3 (Adki & Weitraub 992): Let G = ( G ) for G is closed uder biary operatio ad satisfied ( a b) c = a ( b c) for every called a semigroup a b c G The G is Let H G H is a subgroup of G if it is closed agaist the biary operatio of G ad H itself a group with the biary operatio of G The followig theorem is the simple way to prove a subset of group is a subgroup Theorem (Adki & Weitraub 992): A oempty subset H of a group G is a subgroup if ad oly if wheever a b H the Proof ( ) Suppose H ab H ( ) ab H G ad H subgroup G For ay Suppose for each a b H the ab a b H the b H so that H will have to prove every axioms o group For ay a b H ad b = a so aa = e H (idetity elemet) After that for a e H hold true ea a H = or i other words a is elemet iverse of a Furthermore for ay b H so b H we obtaied ab H (biary operatio) Now we will give some defiitios related to algebraic structure rig which is built from a set ad two biary operatios Before we give the defiitio of rig see the followig defiitio Defiitio 4 (Dummit & Foote 2003): The distributive laws of biary operatios o a set R is for every a b c R satisfy x ( y z) x y x z + = + ad ( ) x + y z = x z + y z 3

4 Jural KALA Vol 9 No Page Defiitio 5 (Adki & Weitraub 992): A rig R = ( R + ) for a oempty set R with two closed biary operatios (additio ad multiplicatio) satisfies ( R + ) as a Abelia group ( R ) as a semigroup ad distributive law A rig with the multiplicatio fulfilled commutative properties is called commutative rig Furthermore if the algebraic structure ( R ) is a Abelia group the ( R + ) is a field Let S be subset of rig R ad closed uder biary operatio of R S is called subrig of rig R if S is itself a rig with operatio from R Defiitio 6 (Adki & Weitraub 992): Let R be a rig ad I subrig of R Subrig I is called ideal if satisfied x y I ad xr I ad rx I for x y I ad r R Let P be a ideal of commutative rig R P is said to be prime if for every a b R ad ab P the a P or b P The explaatio o some defiitios of module ad submodule is give ext Defiitio 7 (Adki & Weitraub 992): Let R be a rig with idetity ad be a Abelia group with additio i Left R-module is a abelia group together with a scalar multiplicatio mappig : R with ( a m) a ( a m) = am that satisfy the followig axioms a ( m + ) = am + a ( a + b) m = am + bm ( ab) m a ( bm) for a b R ad m = ad R m = m ii Right R-module is a abelia group together with a scalar multiplicatio mappig : R with ( a m) a ( a m) = ma that satisfy the followig axioms ( m + ) a = ma + a m ( a + b) = ma + mb m ( ab) ( ) = ma b ad m R = m for a b R ad m Let R be a rig ad is R module a N is said to be a submodule ( R submodule) of if N is subgroup of group that is also a R module usig scalar multiplicatio o Defiitio 8 (Wisbauer): A ( R module) is called Noetheria if ascedig chai coditio hold for submodules of 4

5 Achmad Abdurrazzaq et al 30 odule atrix I this sectio we will discuss about algebraic structure module of matrix Lemma : Let m ad be a positive iteger ad R be commutative rig ( R) together with additio matrix is a Abelia group Proof: Based o Defiitio 2 we will show every axioms of Abelia group for ( R) with A = a B = b C = c ad a b c For every A B C ( R) satisfyig: a The additio operatio is closed for ( R) Because a b R the a b R R A + B = a + b = a + b + so we get that A B ( R) + b The associative holds for matrices additio because etries of the matrices are elemet of commutative rig R so ( ) ( ) ( ) A + B + C = a + b + c = a + b + c = a + b + c ( ) ( ) ( ) ( ) = a b c a b c a b c + + = + + = + + = a + b + c = A + B + C c There is zero matrix E ( R) with etries additio idetity of R such that A+ E = E + A = A d For A ( R) there is A = a ( R) a R is ivers elemet of a such that A A A A E + = + = e The commutative holds for matrices additio because etries of the matrices elemet of commutative rig R so B A A + B = a + b = a + b = b + a = + Because every axioms of group ad commutative holds for matrices additio so ( m ( R) ) + is Abelia group Afterward let ( R + ) be a group ad S is subgroup of ( ) followig coditio for set of matrix ( R) R + the we have the 5

6 Jural KALA Vol 9 No Page Lemma 2: Let S be subgroup of ( R + ) the ( S ) is subgroup of m ( R) Proof For every A B ( S ) with A = a B = b ad a b 6 ( + ) S based o Theorem apply ( ) Suppose ( S ) ( R) ad ( S ) subgroup of m ( R) Because S subgroup of R the A = a ( S ) such that B + A = b + a ( S ) ( ) Suppose that for every A B ( S ) the B A ( S ) ( ) the B B E ( S ) A B S + For ay + = (idetity elemet) For ( ) E B B + = ( S ) Further B ( S ) holds B ( S ) B E S obtaied B A ( S ) + Let R be a commutative rig ad based o Lemma ( R) have the followig result for ( R) Theorem 2: ( R) m betwee matrix ad scalar m is a Abelia group we is R module with the scalar multiplicatio is multiplicatio Proof: Based o Lemma ( R) is Abelia group We will be show m ( R) R module with the scalar multiplicatio for every r : m ( ) m ( ) R R R R ad A ( R) ( ) ; r A ( ) r A = ra is Beforehad we will show that the scalar multiplicatio is well-defied For every ( ) A B R ad r R with A = B apply r A = r a = r b = r B This mea the scalar multiplicatio is well-defied Next we will show every axioms o Defiitio 47 that is for every r r 2 ( ) A B R satisfy r A + B = r a + b = r a + b = r a + b a ( ) ( ) ( ) ( ) = r a b + = r a + rb = r a + rb = r a + r b = r A + r B r + r2 A = r + r 2 a = r + r2 a = r a + r2 a b ( ) ( ) ( ) R ad

7 Achmad Abdurrazzaq et al = r a + r2 a = r a + r 2a = r A + r2 A c ( r r ) A = ( r r ) a = r r a = r r a = r ( r a ) = r ( r A) d R A = Ra = R a = a = A This mea ( R) m is left R module Because R is commutative rig ad the etries of ( R) are elemet of R the ( R) also right R module So prove that ( R) m is R module 40 Noetheria odule atrix Let G be a group G is called Noetheria group if every subgroups of G satisfy the ascedig chai coditio (Zassehaus 969) Accordig to the that statemet we have the followig coditio ( m ) Lemma 3: Let ( R + ) be a Noetheria group the ( R) Proof Suppose that 2 3 m ( R) A A A K are subgroups of ( R) + is Noetheria group m ad for every subgroup of the ascedig chai coditio holds such that A A2 A3 Sice R is Noetheria group the there is subgroup i Ad the there exist j = 23 K such that S j Si Because ( R) theorem m S of R such that A ( S ) = for i = 23 K i i = ad Ai ( S j ) = for ay i j is a R module the based o Lemma 3 we obtaied the followig Theorem 3: Let ( R) be R module if ( R + ) Noetheria group the ( R) Noetheria R module Proof Suppose that 2 3 N N N K are submodules of m ( R) subsets of ( R) ad N N2 3 Abelia group I other word N N2 N3 K are subgroups of m ( R) Lemma 3 we kow that if ( R + ) is Noetheria group the ( R) it meas N N2 N3 K are N K together with operatio of additio matrices is ( m ) ad based o + also Noetheria group So the ascedig chai coditio holds for submodules N N2 N3 K such that N N2 N3 Afterward because ( R) is Noetheria group the there exist submodule N i of m ( R) i j = 23 K Because N N2 N3 K are submodules of m ( R) chai coditio the based o Defiitio 8 we obtaied that ( R) R module So we ca obtaied N j such that N j = Ni for every i j ad ad satisfy the ascedig m is Noetheria 7

8 Jural KALA Vol 9 No Page Before cotiue with the ext theorem otice that for set of matrix ( R) have matrix with the etries is o i j ad 0 elsewhere as follow for each i = 2 K ad j = 2 K m E = 0 elsewhere We will use equatio (4) i order to help i provig the followig lemma Lemma 4: Let ( R) be R module the m ( R) is free R module Proof Based o the equatio (4) we have the followig set of matrix E = E i = 2 K j = 2 K m ( ) { } Clearly ( E) is subset of ( R) ad for every A ( R) m we (4) ca be writte as A = RE + RE2 + RE3K + RE + K + REm Ad the because the etries of E are i i j ad 0 elsewhere such that the equatio RE + RE2 + RE3K + RE + K + RE = 0 oly applies for R = 0 Proved that ( R) is free R module with basis m ( E) Based o Theorem 3 m ( R) Noetheria group coditio The for Noetheria R module ( R) followig fact is Noetheria R module if the Abelia group satisfies Theorem 4: Let ( R) be Noetheria R module the m ( R) R module m m we have the also Artiia Proof Note that based o Lemma 4 ( R) is free R module with basis m ( E) we ca write ( R) as ( R) = RE RE2 RE3K R K RRm We get that ( R) is direct sum of irreducible R submodule so that m ( R) semisimple R module Afterward because m ( R) fiitely may summad of submodules It meas module m ( R) clearly that ( R) is Artiia module Let be R modules ad ( i 23 ) i So is is a Noetheria module so there are has fiite legth so = K submodules of the the direct sum is deoted by 2 3 K implies i ( + 2K + i + i+ + K + ) = 0 ad = K + For Noetheria module ( R) we obtaied the followig theorem related to direct sum Theorem 5: Let R be Noetheria group ad R R2 R3 K Rs R the the homomorphism R R i = 23 K s satisfy of ( ) ( R module) ad ( i ) ( R module) ( ) ( R ) ( R ) K ( R ) ( R) 2 s 8

9 Achmad Abdurrazzaq et al Proof Let R be Noetheria the for every R R2 R3 K Rs R apply ascedig chai f R R R R f A = Tr A coditios Let : ( ) ( 2 ) K ( s ) ( ) ad ( ) ( ) for every A ( R ) ( R2 ) K ( Rs ) Notice that ( R ) ( R2 ) K ( Rs ) equal to diagoal block matrix with the ( R ) ( R2 ) K ( Rs ) as etries of the mai diagoal such that f ( A + B) = Tr ( A + B) = Tr ( A) + Tr ( B) = f ( A) + f ( B) ad f ( ra) = Tr ( ra) = rtr ( A) = rf ( A) for ay A B ( R ) ( R ) K ( R ) 2 s ad r R Next we will show that the homomorphism satisfies the ective ad surjective coditio { } { Tr ( A) 0 ( )} { ( ) ( 2 ) ( ) 0 ( )} m m R m R m R R K s R { ( R R2 K Rs ) 0 ( )} { ( 0 0 0) 0 ( )} R K R { ( 0) ( 0) K ( 0) 0 ( )} { 0 ( ) ( ) ( )} R R R2 Rs ( ) = ( ) ( 2 ) ( ) ( ) 0 m m K m = s ( R ) Ker f A R R R f A ective = = = = = = = = = = = K For every A' ( R) be A ( R ) ( R ) ( R ) K ( R ) such that f ( A) A' that f surjective So f R Noetheria so there will 2 3 s So prove that ( R ) ( R ) ( R ) ( R) K 2 s = So we get I Theorem 5 we used oe of operatio o matrix to defie the homomorphism of module fuctio so we obtaied the followig fact from Theorem 5 f R R K R R ad for every Propositio : Let : ( ) ( 2 ) ( s ) ( ) A ( R ) ( R2 ) K ( Rs ) with f ( A) = Tr ( A) the f ( A + B) = f ( A) + f ( B) f ( AB) = f ( BA) T ( ) = ( ) f A f A Proof: Notice that for A ( R ) ( R2 ) K ( Rs ) the mai diagoal are elemet of set ( R ) ( R2 ) ( Rs ) elsewhere Next for every A B ( R ) ( R ) ( R ) 2 s s s s A is a block matrix with K ad zero matrix K satisfy ( + ) = ( + ) = ( + ) = + = ( ) + ( ) = ( ) + ( ) f A B Tr A B a b a b Tr A Tr B f A f B ii ii ii ii i= i= i= Because trace of matrix meas add up all of the etries o the mai diagoal ad also R R K R is block matrix the the etries besides the mai ( ) ( ) ( ) 2 s diagoal are zero Whe we multiplied matrix AB the mai diagoal have exactly the same with matrix BA so we have the followig result 9

10 Jural KALA Vol 9 No Page t s t t s s f AB Tr AB Tr aiqbqj aiqbqj bjqaqi Tr bjqaqi Tr BA f BA i= j= i= i= j= j= ( ) = ( ) = = = = = ( ) = ( ) Based o proof of poit (b) we kow that trace of matrix is obtaied by addig up all of the etries o the mai diagoal The mai diagoal of a matrix is same as the traspose of the T f A = f A matrix so it is clear that ( ) ( ) 50 Coclusio Let R be commutative rig ad ( R) is set of matrix with the etries are elemet of R I this study the etries of the matrix is the importat factor to solve or to get some coditio or properties of the algebraic structure matrix ( R) I this study we built a algebraic structure from set of matrix ( R) which is module By determiig the etries of the matrix ( R) we obtaied that ( R) ca be a Noetheria module that is the algebraic structure ( R + ) is Noetheria group This study oly determie the term of module matrix ( R) m which ca satisfy the Noetheria module ad some coditios as the result of this Noetheria module matrix ( R) ACKNOWLEDGENT We would like to thak Jural KALA Eterprise Lot 36 Jala Patai Kampug Pegkala aras egabag Telipot 2030 Kuala Tereggau for supportig ad fiacig this research Refereces A Abdurrazzaq Ismail ohd ad Ahmad Kadri J Isomorphism o Noetheria odule atrix AIP Coferece Proceedigs (206); doi: 0063/ Adkis W A & Weitraub S H (992) Algebra: A Approach via odule Theory New York: Spriger-Verlag Aldosary F A & Alfadli A A (202) Left Quasi-Noetheria odules Iteratioal Joural of Algebra 6(26) Ato H & Rorres C (2005) Elemetary Liear Algebra with Applicatios (9th ed) Uited States of America: Joh Wiley & Sos Ic Camero P J (2008) Itroductio to Algebra (2d ed) New York: Oxford Uiversity Press Ic Cuog N T Quy P H & Truog H L (205) O the Idex of Reducibility i Noetheria odules Joural of Pure ad Applied Algebra Dummit D S & Foote R (2003) Abstract Algebra (3rd ed) Uited States of America: Joh Wiley & Sos Ic Fraleigh J B (2003) A First Course i Abstract Algebra (7d ed) New York: Addiso-Wesley Publisig Compay Grillet P A (2007) Abstract Algebra: Graduate Texts i athematics (2d ed) New York: Spriger-Verlag Naghipour R (2005) Ratliff-Rush Closures of Ideals with Respect to a Noetheria odule Joural of Pure ad Applied Algebra Wisbauer R (99) Foudatios of odule ad Rig Theory Readig: Gordo ad Breach Sciece Publishers Zassehaus H (969) O Liear Noetheria Group Joural of Number Theory