Corollary
:
M
bet
-
be
T.fm
generated
finitely
ideal
let
a
be
an
duels that
2
am
of 2
They
ohehthert
n
.
{ AIME / xtaimtmy
AM:b:*
Submodule
.
odette
1^[2-1614]
.
:
M
# a)
RE 1
RM
Prod
=
a
m.ami-M-Q.
t
o
endomorphism
identity
ICM )=M=a#M
be
:
of
an
I:m→M→=¥¥'É-7
R
is
M
✗ c-
-
module
act
R
M
r:
as
-
v. in
m=m_
412
view
=
endomorphism
M
II.
End Cml
→
I
=
In -1 QUI
.
.
.
+ an
=
0
a=÷Ia
%÷:÷→mm
Nakayama
Let
M
and
let
contain
d
of
be
f- $
a
ALAN
in
A
.
-
A- module
radical
Then aM=M
.
last
result
have
F Rt Rsib
NEI Moda and
By
the
nM
RAS
and
NI
aER
.
of A.
ideal
Jacobson
M¥
-4
we
Lemma :
's
lmoda
.
miami
to
RER
it
Is
unit
.
sina.me?o-nI-r,-i.m-iMm-i:IE?- rm-
m
ii. ¥0
.
=
a)
( m)
T.cm)
=
=
0
0
=÷÷É:÷
I !-aaI
÷÷:→
•
=
Ooh )[ 9117 'm]
!
(
Qcñ
=
=
a) [ my
dcñl) @ cmon))
denim
Icm )
=
m
dcn-Hmcmt.tn
ñDcm1=y
¥5k
ñ
:)
Corollary
f- g.
:
M be
A- module
dub module
ideal
be
Let
-
.
of
Then
M=
•
,
N be
let
M
,
a
a
AIR
let
.
am -1N
N=M_
Proof
:
Sa
submodule
consider
As
NI is
a
can
we
M
A- submodule MIN
.
nc-A.n-CMI-NT-nm-t-N.at#=M/N
then
MIN __m%Ñk
M=N_
.
AMH
Peg
-0nA
¥
that
Mcf
R-module
aoubmodukofm
ii
.
:
we
M
→
have natural
MIN
.
Sales
.
which
of Mini
correspondence
are
one-to-one
outs module M
with
in
which
contains
N
ACF)= If
.
,
Lamm
'=Ñ(aHn7
)
))={mM%Ymµg
d- Yacmlw
ACMIN)=
0€
a
RE
(
a.
m
=
done
-1N
)
Green -1
a
)
N
.
( Mlw)
am __=←n¥
n'm -1N
z+N=
↳
⇐
⇐
⇐
.
n
-
n
n 'm
=
at
E
N
'
n'm+n_
am+N_j
a= F.
Recall
the
ring
has
only
ideal
inanimate
A.
local
a
which
one
one
M
that
be
local
its
is
.
and
ring
maniacal ideal
Ahi field
.
Residnefieldsy-posemis.ci
-
A- module
consider the
M which
Consider the
•
outs module of
.
mM_
.
.
R-module
M¥M
.
µ
.
As
of
by
R module
a
elements
get annihilated
-
Mlmm
.
Mi
z -1mm
M
E
Mi
ne
n
(
z
nzt
/ mm
)=m_M_
-1mm
MM
nz
-
=
Kord
→
=
MM
mm
TA-noduk.cl
A
:
c-
.
End ( Mmm
)
{aeA / Qcatg
ME Kord
0
.
endomorphism
Q
NE Ker
@
Cf
Cn
Ca
)
0
=
) ( z -1mm)
=
MM
that
R2 -1m
ft
m
But
M= MM
does not
that
At
what
Q
:
.
.
have
we
A
.
guarantee
Me
Ker Q
ME
M
.
→
End
p¥YÉ
CID
A
-0s
End (
P¥m¥
ÑCx+m[m
[
Fm )
-1mm
)
Qmm)tmM_
ÑPCn) ( -1mm )
=
m
dincm -1mm)
=
=
Need
to
②
dcnlmtmm
prove that
iomelldfined
?⃝
atm-Y-mtqcn-m-ICY.tn
dot
But
=
dcy )
.
ntm=Y -1m
em
sty
pen -47=0
.
sty c- ME keep
don)=dcy )
as
FCN-mj-dcy.im )
had
-Ñ
A- module
we
M1mMAYm-modak_
M
is
/ mm
£1m
be
{
:
local
'
ni
Let
-
sing
over
*
be
let
M
.
.
Kiang
Mlmm
a
their
as
form
neetu
{
Then
M
space
Let
A- module
elements
:
Suen that
as
vector
.
Bo¥im
a
a
in
images
a
of m
basis
Atm
space
Kiang generate
over
.
ni
:
a
R-module
.
.
let
N=
by
{
'
need
submodule
ni :
Kientz
to
prove
N=M_
First
of
☒
generated
We
.
that
.
.
N
M
is
hone
we
following
outs module
a
the
maps
Ni↳M→
MMM
.
A-moduk-mapspoi
.
l
v
Ammi
→
claim :
poi
-
ironiyec-t.me
.
If
this
then
map
.
NtmM=M_
Poi
=
Cf
suppose
for any
F
surjection
is
ne
-
Cn )
ntmm
=
of
omgection
n' + mm c-
MMM
.
that
N such
den)=
.
'
m
-1mm
'
ntmmem -1mm
n
-
m
m
'
=
'
MM
n -1mm
c-
.
MIN-mM=N#
we
are
done
by
last
proposition
.