Monemoduledefmt

If M is an Rmodule Ni M i e I

dine Ni to be Eni InieNoAntesum call t.tkteh

if A CM shutmy o

RA riaiInreR.aieA3CA7fmk.smdlestsubm.d

l M conty A

BI REA Ra if M Ra soca

we say M is cyclic

Def Gen a collation.FRmadsMi ieI

Deke Q Me to beat.ffmtefrn.ls.msEmi

ei fmrtely nay to tuns

Milo t.pk 1 onlytuttyn

to terms

external ductand

cniwsalpmptyitlomp.Q.IMion

fi c IT Homelmi NEI

u Ta fiMi N

Emi 3 S filmits ic

JJ.CIAnte

1211 III Mi hasthestateof an Rm.dk

them rlmila.Ilrmilo.ee

Unirsalpmpty

HomLN.ttMil HIHomlN.Mi

Y IimsHi

Det If Man Rmad Mik M ieIslands

s ohthatE Mi M

independent ie Egmso mi o ski

then we say M.amun.ldneI sum

Mo M

and me get Mi M

Tensarpnaducts

121 M in Mod R Coght Raw

µ in R Mod left Romo

p an AhelmngpCin Ab

A bilmer map g Mx NP

is a map of sets suchthat1 q ma tmz a yCon n tq Man

2 y mini ud ylm.ndtelm.ua

3 qlmn.nl y Minn

DefA product MapN is an Ahelugp

unique opto isam ul folkyuniversal

property

Bil MxN P Hamas MGM P

Dd The Q product of M Nis

MakNsM q12 Free Aldean

gp gematedsolyp gen hyallelmts by thesetof the firm MXNMatMa h M n Man

m n tha Mnd LM na

Cmr n Minn

Hamas MarMP fcHomabKMXN7 PT fLRIo

EFeHonatslMxN.P fTm.t.mun

Fcm nl mr.n

BillMxN Pet g

MarN gen by repsin MXN

notater man g Tn9simpletensions

m ran Maru LMTM n Mi htMza U

et R F field V W R modules

recall 5,7 Vaw P bilm means

Su w tve i LY w ta WD

Assume inged 5,2 is cloud and melt

by Rimage isconked in an R mad

Alvin Edu w u w

f wLu kw7 fue if Reomm

if eishntrV 343basis fr w

it know Lei fj7 xijePthen SEaiei.sbjfj7

fgaibjsei.f.gl Saibjaj

Carmody gren M N P anddint

would loletedledijep

Saiei shjf Sabjajthe mates sniff is

an R onadok

So if RcammyBICM.xMP MnmlP

M R N Rm

x ta in y IIIsay xAy A Cag

AGrammatrx

Redisymm.bihrfmvxvspveohspuesmfddlblu.in

HmvGmm y R

V rub spbluaw ebCwav

Recall stew sym bitw VxV p

yw wcw v

Red Altrulyfrm a Vxv p

alum 0 all utv

Altruly stew symmeto alutuyui wj acyvjiacw.ae

a yw aCwv

stew Syn Alt unless za

w un s w Yu z wait 0

EI 24221 24321

b.hr maps 2422 2932Is P

G za b g la 2b y la bO a

ceca bcecab

i

Glob tacob 9 0

242202,24321 O

2128 ee Q2

ft

iiiiiiii

cair r f Yb at 22