tüte
1.3 Modules
Let R be a commutative unitaryring

Definition 1.33

ARmodule

Mitandaconsists ofscalar multiplication

Rx M
such thatKfor allx

LIIKB

Xxx PXiI x

LXmaxand LIBER we have

BY

KB X
LINYI

M

x y EMIXan abelian group

LYx

Remark 1.34
iIf Risa field R modules are R vectorspacesWe can define R modul homomorphism submodule
quotient modules similar as linear maps subspacesii

and quotient spaces
Every R is a R module over itself its submodule
are exactly the ideals of RiiDefinition 1.35 i let AEM be a subset of03
COan R module M Then
CAdenotes theGranet

VIER die A

submoduleofMgeneratedbyA
itAis called a generating
If A M thenIf there exist a finite A with A M
setofthen Miii

A familyiscalled finitely generatedmelements
mit M
yep ofifcalled linearly independent

ErareMxOX Efwith ry ERKo for all butfin many implies

KO forall XE

A ein independendent gen set is calledinhat

hasIfMabasis then M is calledfree

Remark 1.36iLinear algebraRisa field then every R module is freeIfTheia2 module M

isnot freeAÖ3Kzr.meru

does not generate M

A STIEGT arenot ein independent since 2 F ÖProposition 1.37

Let R be acom

unitary ringR module the following two statements

For

aare

equivalentiEvery submodule of M

1

Any sequence M C Macof M eventually stabilizes
nWe have Mn Mat

Proof AWZ

Mai

is finitely generatedof submodule

i e for some

iDefinition 1.38

A R module which satisfies

the conditions in Prop 1.37 is called
noetherian module
namedone ofaafter

Emmy Noether
A ring is called noetherian if it isa noetherian module over itself

iExamples 1.39
1it Fields arenoetherian

The following ring is not noetherian

fmxxglmez.ge AB

define for nil the ideals submodule sNow

axtMn
wefldez

FIXE QM

RgetM

aCh x

sequencea En MEZ

GKINEXQAof ideals submodule

Mz C

which does not stabilizes

The ring RiOk kz

is fingen

R module But it is not noetherian
is notx x
since the ideal

as afin gen
Proposition 1.40

Let R beanoetherianThen Mand M a R
and only if N is finitely generated
moduleProfringis noetherian if

HWCorollary 1.41
iPIDs

1

Finitely gen modules over PID

Proofiaarenoetherianarenoetherian

Principal ideals are fin generated
follows from Prop 1.40

A2Integrality

Definition 2 I
iAnCANF

algebraicnumberfield K

field extension of Q
Cki e

isafinite

dimake aandThe elements of karecalled algebraicnumbers

iA number xek
anof a ANF is called
algebraic integer if it is theZero of a monic polynomial withInteger coefficients i eaforsomeax
ananOERWe set

EXEKI x als integer

Thisiscalled

the ring of integers

of kFrom the definition it is not clear that OKis actually a ring To see thiswewill
study the notion of integrality in general

Definition 2.2

Let ASB be an extensionof rings An element be B

is called

integra or

if it satisfies

aa X

an 0121

Thecalled

X twith coefficients dieA

Integral over A if allover A

monicequation

Bisringelements
be B

are integral

We want to show that sums and products
of integral elements are again integral
Proposition 2.3 Elements bin be B are allintegral over A if and only if Afb _b
is finitely generated as a A module

To prove thisfromlinear

recall the following fact

wealgebra
Laplace expansion

Proposition 2.4 Let A
with entries in a ring R

be the adjoint matrixtidiwherethebeai

an Carl matrix

and let Ai e

det Aiis obtained from A by deletingi th column and j throw

AifAThenaAA

AHA det A E

In particularforAxOXERdeweEmigetXO

Proof See linear algebra and notice that
it works for rings and not just fields AProof of Proposition 2.3
Let b be integral over A

for someSincemonic polynomial

f ismonicgate AGanyfbOf x e Af of degree
canwritemasg flxgwei e

trafor some qu rede AGwith degenen

Therefore
rlbg bam b

die AwithAfD

If

dot abt

bebnis generated by lib
areinduction on ngeneratedb

integral over A we see bythat Afbh is finitely

assume that Afb b

Converselyb Wi

det BEisinceIC W twrite
Gi EAStadwegetSwih.IT
LEAfb

Moreovercan11111

By Proposition 2.4
for allwedi Wja

Then forWr

is fingen by aany be Alb c bA0b

we can writeEwr

with Ci EA

detlbt sOtr

FicifiSinceSb

we obtain

thatdet b Ea be

die A

anbis integral

overA
AIf b and be are integral
A then bebe and be be are also

Corollary 2.5
overintegral over A

Definition letiThe

EEG be B b integral

is called thei

Ac B bearing extension
overAintegral closure of A in BIf Ä A then A is called integral

closedin B

In particular

If Kis an ANF

OK is the integral closure ofZ

thenZink

Proposition 2.7 Let A EBEL be tworing extensions If Bis integral over A and C
is integral over B then C is integral over A