MKovaov -

St 9, 220

S G ana m nahan GxG G( ulei caton)

unit ela t G , '8*33 eG, asso a Ehagh

MAC Crs

de Ta aM iek X. Ta s mndnes Sy (x .a

easie do udston h.

Exo-ple ee vs. issaks equialnaR

ho SmC2

C2- -2a cos Smp mas a9 anu >60

Sn smmties n-anass st

R S ies (innsh lua maps) veakn spa

2)GLn (1R)

Dha&al n D. 2a aks -smes f lau

S)On)- Sxm es nScaR

4) S

Exed: )R«I- an kjest

CAR C

anSunS. OnRp

us exenciu as ma

P Pprir)

Fie umk fuo

tra rape hen g& is a prods ma PCies.

Pna Aan a ch easin unndan on ass a aki

os >C,,*a2 .

-27 Ma ma have te dpns asic dw eas

IN-l,, 2,. . natl umas +

Qo&or amns IR a wnlnt, Colax mmlens

A+ B, A B M(IR) nrn mahias nAmadias

De A R-(R,4, ) isa ct Rum tho bina

sach dlat

(1 R,t) is en a ar groP ,

. a don + tmns R

(2) Onado aSSoave. Sh ab Lnsttas a b

(3) (ndnackon axioms RAn t a. . ht ons iht &shiuhue laws ld

(atb)c = act bc cla+b) Ca tcb.

(1) atlh=bta , ab) tC =at(btc) o ia6h (R,+)

a dihe iasy - a aie nvan af a +a =atO <a

ael-a) =(-a) +a =0

(a) (ab) c= a(be). Usuay 4iu t ptha (2)

a eR Aa af- a

Calld. wgs We ms st aaS onsi 1, aSSu

enhk kr +

Basic Praies O 0+0 O. a = (O+0>a >

O.a O.arOa Oa OarOa aaliar saup sdan t

bth=bho

Oaso

) Oa= aO = O a eR uliacadon 0 s0

-a ab a(-b)

- (a b)= -a

3 a swh's q uabon ?

ta us Precadlanca oven -/t

Ob=la l-a) )b = ab +(-adb ,

ans ob =0

Jasthihubi O= a-a)

a 8 hue vensp

O abt (-a)b > (-a) - - ab

a, a-b) - a Can

Ya,beR 2 -a)b = al-b)--al (-a) C-5)= ab

oed

(-()l Pro u. -2a = (-a)t(a)

na at at.. +a 3aat ata 0.aO

na - (na) = -(atat ta -2a -a -a. aa 2-a =ata

na rodtun ua to Ha

(eR ItleR aee 2eR, t. ER de aeR

s

dskagush hala,

-IER - =(-)n =(-t}+-t)T t-)ER nRade'.

MCe map dmes

ouo is pro duct R La mP

i, tive 0t ta (ifDo)

teud suwes a) t(a)t. t (a) tnco

N , , IR, , 2adan maduus 4-

M. IR)

s C

Mu ador in N, 2,, R, C ism u ta e

in Ma (R AB BA Ar an madnias , usua (n>)

R R is calkd

No

Des2 COm da ve T alh ba Ya,beR

oa Com nta tive ab#ba Ror Soma a, eR.

Po posi.) Z,, R, C Comnda te u a

2) MaCIR) Is a nam nunba hue , Br n>l,

Gtnnalis isd hude lawsowelr

prduîs Lmta unluss

s onudahie

intach

Ca tb)Crd) = lar5) c + (ar) d = ac r-lbctadtb&

Biu ehon ca

CatT. +0a)lh,* bt. +5) = 4, ra,h,t. to,bat ahyt@ab. 1 Q.6

(o)E;)- 20b Cannts

(o+6)latb) = aa+ abt ba + ob a tab+ba +R

aR a-a a o=a.a a..

e O o a a.. a saSsoda e

ell-dafwd

Cab)"= ababah.. ah lors, sm

R s o tae Can

(ab) atab ba tb a* 2abtb (a+b(a'rab + be tl2) (a«b) = ata+aba+ab + ba-babt bar

Ris snei 3

a3ab+ 3a b

Po R Ris Gmad and,b eR un Peo - bino i ofhcdt

Cab) ()a*"

l(n-u) Inskaun inducde ihsn

R (A)= (")+ Ca-') Pasca tia

n=l-2.3.. . so tns

Cann avid R =Z, r isa au

2

eaheam is an 3 3

Inde a malus 4 64

Slo to S

(ab)-abab. .ab = a^b"

R R

Co-b)=la-l.) Ca-)= as-a- oa + = a- 2 ab+L?

Tn e sids mdulo n -6-

Ha n 20 an onSi o an equvana laXa ton on

ab na -b

CL a, a ba an b~c

Evalena Sases Rapuga es o, 1,.. n-i OtnZ, taz n-tn Z

a dihon ans us pli a don 1 2 Jescends do a ell-dak naed a9 n ond MuOApkd a on on e w na Sostes

(atn2)+(btn 2) = atben 7Z

ell-des, (2/n, +) isa r-p on elu n.

2)lata 7)(bta z) = abtn Z

needdo ad psdatues

alanu Ao ss a'=a * kn : b +en

aen2=atn2, b'tnZ = btn2 . a a'bZ = abtnZ

a'b=la tn ) (b . en) zab + (al rkl+ke)n. Ge a trdue (Z/%, t, . ) usiluus ml n

s don Ro Cen e chack u Tc aKom3 Zisa

xa~k n-6 Zls = o, 1, 2, 31,s] 3SE2 (nl 6) Ou ds r+k 3+SP2 3-S IS3 (~l c)

3 C 3 in 6. 23 =0

Zeo y 10 in R en a Oa a eR

a =a aom) O-a =0 (eve

.a -O aO a=O Ris a si lawt R o}

so nonphism sA mnpe - isan isononpism R

s a bijechan sets

Ra ttuns

atb )= Jla)t (6) arU a, R

ab) = 1la) (&) ¥a,b eR

muln R ult n S.

Canso urit ipe

r R Exench aAn isomopisn leR do le$

L (o) =O 1r inS.

SR He nve od ,also on iso msrpsn

Copsin isoess S an ophaS m

Gentna Sahabon Ar isanopis jekh is ^ijrkon A srds

Soepis m eds (have sa candina G)

so pis ven spaas (hae so dari)

snws oS.

-8 Gnven a ,a sulun SeR ic a su el sudL Ot is an alon snant R

(2 S is sud undn nulhi a dio n So mes oni on (3)

drop aJ (î uully

ExamouS tan Tts on

7 IR CC

) sulus

L R

2) RC M (R) ala eR tor ()

aaeR

to Z. t e { JDL.-4

So oa.

Su, m EZ

n no 4iq

n

all a Dr mas

Exen u 2 [) is 5u PCm tory

20Pow Can

n=pP*? 2) Wan i 24)-zLt)? 3) Does hae sdmE hgoas foe ue ualy le

,2, Z[4),a?