i FQQI/LOFL Xl open la) : (i) Commutabve lag ith Un tg - & This a cing Eth bes HS mur pha kien ofecation to be Commut ahve and hat a & mo ltspiicative identity. (Gi) Bee com motanve lag eth Opitz? Thi SA 0495 Lye hes HS meltifire ates Oferator +o be Pontummvtatve ond has a roo (tif liebe VMentity, (Hi) Sew feu: Tris i & fing wrh Ident 7 ihe hy, = every Nenzao le meat hat a moltifliative = inverfe. LATA WAH TH x without indi Exemple of Comnctetive cing eerie Unity : (ZA +.) (RAY, +9 Gi) Example of agtommvotatve cin ath: ones / See fA; (24) PaecdeRy Wh “mattin additen ond meth)e ™ iti Plicatien, KALANAUNUHOT Scanned with CamScannerCongratulations!!! 3Craduation g Prom high-school & le) Zi ij tre See yf veges a aA® &= aib- | a a@ b= ab - Gto) +2 a frog of Rimy fapertiel i) Closore under G): _ Lee ~bLEZ. s e@® b= asib-' EF , Sue a ple? s é & 01) Avot iatnry vt ©. gt let iw bine eo. ' . F a@b = atb—-! be = Etec -I a 42 GeeSsO. x + Ca@b@c = G@to-l+ Cc —! 9 = adbtc-2 on “*@Ocb@e) = a+ cures—i)-! 3 = atbt¢ =2 : Topelola, Abidakur, ——— Friday, 1st October, 2021 Scanned with CamScanner2+ @@b0®¢ = a@eec) OD) Commuteruiy of @: ~@es atb —! b6@az= bta-'! = aAtb—-! -aAa®keh = boa av) Tdetity e: LeSome @ 13 te identty of @- eo AMC EA a ate-loa = C= nr-at! = ae} “1 iy the addave ideaney . GQ) Ddlixrve inverse of © Lor megtre emery / i Ita ios Keer _ ler a be t) acl Ane jabege Gi Gin ee A RT A ST TL a@® a £€ #4 = atali—le= | = aoa ltl -G => 22-4 VaAEZ, jnverste vf ao Sb 2 oe ce Alte Scanned with CamScannerwn Ey ak Prom high school | ete VW) Closure of OQ: ager abe Z pts ADL = ob - Cov)+2 EZ. Smee Aposhd eZ CM) QSSociomity oF CD? Fea bot a bce ZZ. IO Y= XI -—Cutg)+2 aQdb= ab ~Ga4h)+2 LOC > bt —Cbtc)t2d (PAG) b) CC =(%b —(atbJ4 LC =[ ab - cater + o3 4h zAabl-ac-bl+2e “Ab tatlh —2 -C FF) 7 inh Ih I, LT sabl—ac —bC —aytatetc HE ga 4OCLOO= ag be-cetat) ca —fA+ C6 Mp fab LA ia Friday, 1st October, 2021 c Scanned with CamScannerfe Vesa. ~ab-al4tin -AaA =-bl tb + Cy +2 : sab - Gb—- aC—be Fath FC b- We See that: Ca@ or = adCbOc) nr fe rr § Oi) fear of © AOQLtz ab ~ (a4)42 = ba —(Cbtadte = bO% ‘abe bO% OW Ermiteore oF Unity /- Let 9 Fe p- esant wuts - 1% e@—O9= @ ‘ => 49 = Cat9)42r=2=Aq j => AG—H4-GHAZDIA oe ae et => AG -9g> Lad = glad =rca-O => 92 >le~\) oo 922 LeThe matt Piorne ibenety/oaity ID, 7 (2:06.80) DO Gan otechve ring its Scanned with CamScannerIMD MR 4 (24 Da, bred eR Rong Peo gerries ber a(t or) 1 OCS a) C= (ae) & MR a Odsere oF +? é Ate: a ee i) - Fen by ba = [8% +o 4, Thr GAL ee where ath, » Artbr, Agdhy 2% 7 b4 (i Ats eS My IR Gi) Comm utr vity of t! A4tb: [o, rag p fbi ic ok bz; bn = [FAithr Asth \~ (Ot br te a,+ b; | by td ba tw, 1. ait pee BFA. by ba Ay An i el Scanned with CamScannerMN Eve high hod, (i) hier tf ef ts (hs gte= He hiss thi dd Ls gq z C } +S 03 4b, )4¢ G (a. tha) tG. = 6, Hb, 46) 4, +i 7G) of os lie a at af Gy Ay | b:46, $44) a os 7 Lb2tG Sbidigd - bi ts fe a fc * auc 2 fA—_+- C4 77 ‘(Atb)+¢ 2 A+ (Bc) Topelala, Abidalun, Friday, 161 October, 2021 ee Scanned with CamScannervy Ervdterce GE addtive Identity: Le = Q On re(rese + the ¢ ae A E c a: ot additive identto. ARIE =A (3 aa) t (‘ %) = ka | eC, Cx Gy Ga _ G, FR, A240, afte & - @ 4Q On a é a “ => aA +t 2°, = 2, 54, °4,2 Ay 4.0, FA, > 2%. -G,=06 Mites 203 = 2259 -& =6 Out Oy FO => Os Gu -A,2 0 ° toi ‘ . Bl a te Gdditive 120 ant (V7 Bxcatence ot trdditve averse; tee fi’ Le tre Mvede of A, Whee / , A’ i Gs ay ¢ 4 ay Ay 5 (melas e(22) ot as Scanned with CamScannerat mn ony Sy a ap as ay Ca CA GRY GY GEERT y, wu, Biown high school —— obi aA, rar =o > aAf=-a, DB 4, +a! +0 as af = -% a, + A co D> Af >-4%, Gu ta ze oo AL = -Gy deep f.~4 623A i the inverp i ~a; =a) ee An A in (MR, ) Clasore. of XY. AxhR= AB -/% 4% )\/6, 4, | A; sl fo 2 [Gs by + Oe Das Abbe On by + Fe, by A, one éE MR Sincg Cech of HS Clements are Peel prey Abidaku.. ——— Friday, 1st October, 2021 Scanned with CamScannerO31) Piocmtinty of & (AB)¢ - a, Ae ey oe ech Here =f bt Fby Ar by + Mabe ve ie Aybr + Mba} lc 6. a) & SCG 614 Mb) 4 G (Aybr Feb) ( C4 gb) Fn bs) £0y (Az br tu bu) CCA, b1 42 by) 9 (Gs ba taba) Ca Cag bey + Gee bg) ty (4g bo tau ba) Bln at @) Coyre x a(cte one oma ee ( By br FO Fay 4 Gash 4 GAqbH Corby + Gy by HAMEL FG Ae by Cry by + CaGe by 1 CMs ba + Cea 2 (“* G +62.G) + Oy Cbs + ba) Oy Chi $brG) + 4% C by G + bu G) i Coils bra) + arbre thn in dH) A Fb tit Ned. Gas iat. Cabiel Friday, 1st Oct: Scanned with CamScannerI y sneer 3 Fam pha —-. i biG +b.G br G thicy \ F fe, aa by a beG + ba Gu) Go Gey a BA; G, 3 2 ACRO 2 na)c > ACBC) E G@ fh R if fot aa nteg fal dorm etn I oc sre(d beca vfe the molt he ant, : Sfetation 1S nek com mutetive. So MAR UW por a Commutoatte L1G + [te en while Integtnl domeial each fre ldS ace tom , _metnt've ngs. Abidakun ——— Friday, 1st October, 2021 Scanned with CamScannerBe DG) 2eto Ditet: Ler Ro be % Commutetve O49; a zero divilel, a eft, JS a Non -2eF6 le mene Sth thet f JbotecR + arth =o whee OO js the add itve ideatity. a) Units Lee Age a rmg with onity. A ont oe ring fh is AN element HER pocn thet phoe enists oe Ro with: ab> ba =| whefe | is varty, an Sobeia a: Lee R we aw cing, A pone Om pty Se bseeS ok Ris Seid to be & Subsing vf Rie ¢ itsolk tS ow a9 the the in deved ofera tions. usder av) Idea : A Sun cinsy A fh a tng RS ceMled oa Idec cf fL if for every re (Lead every ack. woth CA ord ac ae in A. Scanned with CamScannerfy 2>_ Cato) la tb) = at ar t+abthat+L*s ate S@radualion 3y, Fiore high-school 2) a o* a , GQ4+ayr + Ode a a> att attaryarte ata s Be ate +Q #4 = Ate Sin le Gq a = (at a) tata) = @2@ta) +o 5 =o CAataj) 20 by acco (eto dbx 5 t2- Ata Se 3 Boob) cee x ; 1. Wase € Os, é Grp) o4b } =2 a Coe B85 1 chi Erde, thath = ate Jiinte, doz? 2 abt 6a =0 ab = -Cba) 2D Ab = C-bDa Abidoku. ——— Friday, 1st October, 2021 2 Scanned with CamScannerWe heve Proven alresdy har, 7h GS th. bth z0e b> b= -& Ab = (-6)6 sz abibe r. QR iS Com mut ove. ‘ i . Oe 2 Ma IG My eR| : 1.) MiR= (x. ad T{(p aye | Prot ’ 5 i nreltty, 5 ft, T if monem(ty » sace RS torent. i -/% ) ET a 2f% a» , Be . rose (SA) ECE) (-8) = /-b, ms) E eo obey e ale, “| £ ) a, a ial ets be iz t 2 f/G,-b, Ar~be a i ~ ) Ey i o Gu - bn (Vean ob A ome Goen GIR Scanned with CamScannerpeice Sonn SS 4 °: so} fi Hy -/ 4,b,t¢ Q,b.4@:b4 oto ot a4 ba = Gib, —— Ee L _ oO Ag, be pnce 416,» W& bs tba | Gy ba © RR, |] is a fob Org of Mn. 245) Wo went +t Prove | Th a isa unt inZm then G-end om Ate Clatively Prine 1 Tf a Aad m ero celntrely pane, thy TMopelala. Abidakun Friday, 1st October, 2021 Scanned with CamScannerNijeme a ho avat on 2a 3B béZ, G-bil (mod ™) 22 ab —-l zo mn , where XED y Ab- mM =] =p Qed OC SM) =z! zs) ao anh ~m are relatvel’y fone Grvese Imp Wiiater Assure aang M ae felanvely from. Bx, de Ri an tmg= | Px An -l 21- “MY ss AN Dl Cmed ™) Now, grace mE Bb and WIE ZIPE2, YS Pomes Then ax ¢ 27 weed aPareaag Ax 249 2 (| Cmed m) a ih a vart ot 2B Scanned with CamScannervbliputulations!!! Graduation 1y Biowe high school ——______— Zdii) pu tre ower on Z 2 haya py suck are : fp 23,4,5,6. Au tee zero divisors in Zp__are_: nore FF rh rn tel the! All_the units in 25 > {osla du 5,6, 24 are. 1, 345) 7. the 2ere dived in Ze are 2, 4,6. be) a Se) ASE CR,+,¥) aad SECS +. *) be cing S, A cing bememecphiseng, from R inte ws Mmalfiag fren R inte Suoth that: 7 _ Vor be R tla ths fete tu FCaxh) =ftay *’ For) where Fcoy, FWE S. Abidakuy ——— Friday, 1st October, 2021 ie art tie a — i — i Scanned with CamScannerzo) Y R =e y @ 079 hu mom of Pligg () a,6€ ker p » €é R now, kee Y= f re R* Poo = Os} hemoener Now , Yea +b) = Yoa + Cu) * roeies 1 =O 40. Lhe Ub ke Gi 20 f Toate & kerY E E Also, Pca-b) 2 Qu) + Os) é -Ylo + (- gs) t rr _ o+(-o) =6te : =z06 E c t+ awe ke Y = E ans, YCrar 2 Qed ea) = Mr) 0 ZSinc@ a EKO =z O tra € ker & i neo Scanned with CamScannerCongratulations!!! Frown high school ss Ci) ‘to Prove: An homomorp hiro Va Ga L WomerPhism jf aad oly Se Ke (:40} _and PCR) = $ fore i WE ics Sm pligates 4 ASome ¥ if an Some rPphiir Fam RS 3 Then. fofely » Sin ce ¥y 1§ uate 4 Yr: ¢ 3 We prove b+ Conttad E27 SAyene atoe ker GY. Des, Ota-o ig 4 1. Since WO is one - te -one ; ao) > Ole) =0 a. CZ @ a => aA=a90 pobich Corntradiy the a aSiem feren thet ao DB Y_peing 4a Memorfys an inflies: Dy ker Qzfey end PCR) = S. *——— J "ppelola, Abidokur, ph es Friday, 1st October, 2021 Scanned with CamScanner» Con verse Afume Kerp= toh and CRI = S. mou -Siace YR) > S, tren QO is ate, Now ,atl we have tet JS te Soon that js one- Go~ene. Row, (ee abe, Diune Qo) = Os) Wea) -PCb) xe ay Y(a-4) = 0 rSince Ker PHdob, the a-b=e a> az / a Y 1) ene — b-070 i 2? Y {iS aa is ome fhisa. Scanned with CamScannerC MN From high school ee BC) Mg FR PS 1 ae beg seme [we purt te Show thet fi iS blo an ~ isomer fife, ft = Ses FS bivective 404 homomePhic, Fe R SS ta Sot r Let S. Se § J r7E Ri Sz Fe) Sz FVD rc ‘ £'¢ sto x = FCF Ce + FOC) 2 P'CFCrtey)) te an i. ~~ +c’ a 2 FCS) + FOS 4 gcse) = FC + FUG. D : Abidakuy ——— Friday, 1st October, 2021 Scanned with CamScannerPio, F'C S50 =F CFO - For) > FC Feed) 1 te Fan =e. ¢7% > FW FCs) oF LF UCS SD FOOD FCS) re Fo gf homo mor Pic. + To Prove one- te — 078: het Ff Care F') ,aeeS => FCF a) =P CF CH) za asl H or! as Orpe -t-- 9% 2 To Prove orf: Wre R jose £4 FerI= Ss = FEO) > fc aD rif ep => — 1) onto fo Uo [Somerfhic. Scanned with CamScannerara (acd a) We want +t shew tra & YU if home- Let a,b € 7, 1 YCat” = a+b Cameoda) = 4, (med a) 6 2 ba (med a) G4+tbo>Gn F¢ b& (red) Abidaku, ——— Friday, 1st October, 2021 ~_ Scanned with CamScanner, Y lade) = Ath Cmeda) > an +b, = Wa) +Qly 1 Now, Yla) - abCmed a) (et A = Gy (mod a) b= b, Caod A?) A -G, > Ne _*é€Z b -b, = NF ~ce? 1 aA = net Be b =ne ¥b Le abs Cat $a.) CAL tbe) wae THEE HOB, HNC Ae than sy Ab7~ Gaba = an Cate tth. +) Dy Abz= Ay lon Cured a). 1. Glob) a2 Cm a) = Gy, bn 2 Pla) Qo) Diy ha memerdhic. Scanned with CamScannerThe kernel of @ is diatonic fees AE O Cmeodna) = A tnt te Zo kee Q= {nt “te ZY tne vs all pultifles of 0. ia) G) Martimnat Idea ts D menima! idgal DA of A Commute— te _ cing Ais a proper sdert oF fh seep achat wherever (2 i en Ident of K Gnd DP EBEPR » tom g2A oc B= GD) Pome Tdow! * Pe Prme leet J of a Commetanve fag » (me prper Heat of 2 such that so be R peas be A, BD aeA or beg : Friday, 1st October, 2021 Scanned with CamScannerCit MAancifal Tdeat; A prncipel fdeat Pps oy Ris eo ideal generated by & fagle element 2ER theosgh rel 6 Pl: cova by Cie a Clemat of QR. Tt of eitten 9 form: Kay efra sre Rh 2 mee ack 4b) Scanned with CamScannerabit | aw OB eo he baa Topelola, Abidakun, Friday, 1st October, 2021 Scanned with CamScannerhE) We cong te Show that FZ 4 maninol in 2B, Metred [ Sopse 77 EB ce Adjume tat 72 AL cad oe a 2 LAT , for sme 2E LB => Qed Cb. 7)>' bx» 479 =!) CFE s7 J ~9Ee 2 Sice bE end TE they bxeB. and 72E 6, (re @) Web since B iS an seed, }obxn379 €B ,1€8 yre BD eee re rt eB, Sace Bo an Weal, ree 2 az Z. Conttadic pea ! - 72-8 ef Bw YL IF fh BR Ma IMal ado Gl ot ft Scanned with CamScannerv "2 MN ih the wie yeas nn high scoot Methe gree ws 5 Be: Reo method TI ae Aa wa (ae d a ait ee 2 gia on he Ee cove pag 6 b eS bp es oe Nd | es ve fin ce GQ prime av~ab ea A EB So) Gd Princitat deo! Dom aia: es The Poarcipal flea Sic nm ee Dn US wo mn rah tegra) domtsia (0 3D ssnich eVery des! e= Wo 4p forircifel evel es es eee Ongve facterizerta Domain: =— Pia_integri domam Db a Lngve fer te ization domain if (red? Pex 1D every orzerd Cléma+t 9F DP Tete ee a bidakun, <*2—“< =" a. ae ee Pre Fiiday, 1st October, ged ecielel_of D. deawe 2.. Scanned with CamScannerInte sereducbles SS UA 2) Vee factors zanin up te a5foontes wad Facto abfeac, the uederin UK the foly em in af degree Shy) The iredecivle Me and % ta. Pylon) foo has re tere deck le ever Re Con! URE Pes foljnomin't zere fa 2) . fre Juv Vs Die psec UD Ieredvableys lee Ff, OV? Zan PEC SSNS & a #6 , @¥e fo win Segree ofa. nsve ah | coctth» has CR Cc #o . ase mor- oO, bef Be = do s+ The PoKGivie Polgnomrely are xe tin ey Oe obey ts yr FOm F! See | | | > iSo5 17 yt Bk ES (modb) Mrtl. pushen 2% +1 Le yhat Hb feducrble + ne POO Cor Gaie 6 SES ae boee weeks aa irtat alee Scanned with CamScannera we scliool . eee ee ay accee 8 Fredeupies wes % 3 iE 3 we Lo fpr) = are + be +H F wa Gee 6 d#e Pihimvtln tS “Teefens « 2. The posible folgnemr ials afe: Pome + ett : eds etext 20 + Wp + Ontt * asqrere re + one + ett! nF tree 28 + oxt + ost! ?} 20 +1 When esi, WE EX HAT! = Br oO Cred 2 wher zt, 233) > 2 26 (rod 2) | xteorti £6 eeu boty 91 250596 =-) J re +1 2G foc bot pea % =I Cr IH! aed. ee Hl are ne Poreduciblesy of degree 2 in Zee]: Friday, 1st October, 2021 Scanned with CamScannere ba A Roa Sta a fel naae ele lous we “eo — Ga) ber E be a freld. Pp noncwerstant Be felyrominl Fw & FO) 2B KA & be Be _icreduc ble if Hen) ance 6@ ert presied 6S a FH _paodect of tue penwosmat polynemials of loner dea cot thon te degree of FOU. } bb) We ukat t frve thet: fe fF Uo Feld, 46 F ons Fo9e AO ee a are of FOV IF ond cola: See SG iS Qa fut eof FLY. foot Tetlia hon TNT 1 AJSJueme a ECS Zem_ eof Fw) Friday, 1st October, 2021 ARBRE e if Scanned with CamScannerNow 7 By me 2 tou Foye emnIaGd + Cc? where Bepeed c.2 386 DA Biter: Ce te7 Antanas Sa. Sf Ce aS te: 1 fod 2Ol-e)Gew + ¢ DIdEs J ACS EA, WS a eer of FOw thin ; fe) 6s m&-63200+0C = 9 = 6.6 BOVE FESO =>. G52 28. ts Fo See DATS 2 PQO = Ct 95.02) Div tS 164 A190 ith, Cl) ED: deg ror & in FORD == wyether with zesre form on iol <> esi of Fol. as Ansar =. Scanned with CamScanner