chl2 ICModdes.ir D

LasttmeelfRapiD Mc R M is free

afrank En

Mantha If R is a PIDM a Ag R

module

then it is a finite product Gdirectand d

cycle Rmodules

Recalls if Man R mad Mcyckif M Ra

save a cM

s naik Krul

Inthe can get RM

K Rd ok ME RkoRd

what onewould like canonical decomposition

intocyclic

e g Ri Z M 21 21 2 0 248 292 249

CRT 2 21 245 24 x24g A242 24

urge expressions elearnkydensers

coptoaearddqpnmmfrinhrmedodeokpzp.ir tram

2 2 418 48 xf24aJxlD.net

primary components

primarydecomposition

Invarautfactrdecampasihui

21 21 21 9 x4azx x

a lad IanP8,915

CTau I affrdmonlyS highestleft

an 2z

91 2 92 8 93 89.5

Mcexzdxkklxlksdxklg.ae

proaf.FMthmMf.g.RmaduhDefeTarlM

EmoM nm 0 some r RlE

Canchoosegemsts m imeem

Re M

Cri re7 Enimi

sometrue KcRl K free Irked

Claimi Canfind abasis ein eeaf Re

suchthat qq.ae oases basis K see

Sone ai e R

Matei if daintne dare smeeMNRYK.IRqxRax xRlasxRxw.xRl s

aa Rl 124k 17capes f R

Strategy find elements of K whnh are minimally

divisible Kc Rl

Cardsfans Home12412

E qlkllkek.geHomie D OR

let a E be agonad go k te Hom 9127

Set IT g a

by caustaton alpl all q cHome112114

E c K

wile y E aieieikasfrRstmddaiI.ly a i cardfundus in Hoee's

ai dia some die R

can divide and get x E drei ax y

Claimi x canbeextended to abasis

Consider q Re R

y a

x 27

Claim Re Rx Keng

rxtk Lx k

v sfylul.x.lv yard

claim K Ryallarglerustle Cry k

v sleaf y.lu Hyfuk tirdahn

InsummaryKarl found yeaxo ks.chhotRleRxoPLP kryk Rya lPnk rkP e I

contre w Pc Re HannP fun

Prk sk

Me Rx Rx RlainxRlaia xBamm

ai inna's in RCRT

mangy fats can getexposur

Me Rx RX 146 xR zx XNbb Ibd Ibn

Maedekili

PYAR PYa.mx Rahim

a ai al aman ai inned OED

aini aji BR

bI ai s bIg'sai ajar 5 2uptounits

Unrguenessofpnesewth

Suppose Rx Rx Rfa x Ran M

F fnac R Mark

Marte ape Rape 10449.927

014 95

Rape F

1 12917Tat rabbit

of bab tO

Mapf is an F weekspedin atR's in

decomposition

rankof M

If R aR M consider MqRlpRpinned

RKRNRtp.pt Kpmnnr

Rlpii a RhpR0 ifCpips I

RlpR else

kpi.pl I Cpip I

apinitbp I

RlpiniaBp T a 5 Fox 15 F aCapinitbp 5

Fa apinistratops

ofpinionas tr abp5O O

RIpnp.TKpR T4pR

aah s ab

Laa c a

PYAR94247R n speartypeofdin I ilpla0 else

consul pM