Publicacions
Ma emá iques,
Vol
36
(1992),
1011-1027
.
A
bs ac
ON
CERTAIN
CLASSES
OF
MODULES
K
.
VARADARAJAN
*
Dedica ed
o
he
me7no y
o
Pe e
Menal
Le
X
be
any
class
o
R-modules
con aining
0
and
closed
unde
iso no pllic
images
.
Wi h
any
such
X
we
associa e
h ee
classes
FX,
FX
and
¿5X
.
7
.'lie
s udy
o
some
o
he
closu e
p ope ies
o
h ee
classes
allows
Lis
o
ob ain
cha ac e iza ion
o
A inian
modules
dualizing
esul s
o
Cha e s
.
The
heo y
o
Dual
Goldie
dimension
as
de eloped
by
he
au ho
in
some
o
his
ea lie
wo k
plays
a
c ucial
ole
in
he
p esen
pape
.
In oduc ion
Th oughou
his
pape
all
he
ings
R
we
conside
will
be
associa i e
wi h
an
iden i y
elemen
1R
,-E
0
.
Unless
o he wise
men ioned
all
he
no ions
such
as
a inianness,
noe he ianness
will
be
le
sided
whenwe
deal
wi h
a
ing
R
.
The
modules
we
conside
will
all
be
uni al
le
modules
.
In
ing
heo y
he e
a e sco es
o esul s
dealing
wi h
he
s uc u e
o a
ing
R
( esp
.
o
a
module
M)
assuming
ce ain
classes o
modules
(associa ed
o
M)
posses
ce ain
p ope ies
and
ice e sa
.
The
esul s in
he
p esen
pape
a e
o
a
simila
na u e
and
a e
an
ou come
o esul s
p o ed
in
[1], [2],
[3], [4],
[5]
;
[6], [8]
and
[9]
.
In
[1]
among
o he
esul s
A
.
W
.
Cha e s
p o es
he ollowing
:
(i)
R
is
noe he ian
i
and
only
i
e e y
cyclic
R-module
is
a di ec
sum
o
a
p ojec i e
module
and
o
a
noe he ian
module
.
(ii)
Ci en an
o dinal
ce,
i
e e y
cyclic
R-module
is
a
di ec
sum
o
a
p ojec i e
R-module
and
an
R-module
o
K ull
dimension
<_
a,
hen
he
le
R-module
R
has
K ull
dimension
<
a
+
1
.
*While
ca ying
ou
his
esea cli
he
au ho
was
isi ing
he
Ta a
Ins i u e
o
Fun-
damen al
Resea ch on
in i a ion
om
he
Na ional
Boa d
o
Highe
Ma hema ics
o
India
.
Also
pa
o his
esea ch
was
ca ied
ou
a
S an o d
Uni e si y
whe e
he
au ho
spen
a
po ion
o
his
Sabba ical
lea e
.
Pa ial
suppo om
NSERC
g an
A
8225
is
g a e ully
acknowledged
.
101
2

K
.
VARADARAJAN
In
[4]
P
.
F
.
Smi h,
Din
Van
Huynh
and Nguyen
V
.
Dung
gen-
e alize
hese
esul s o
Cha e s
o
module
heo e ic
se
up
.
Le
X
be any
class
o
R-modules
closed
unde
iso no phic
images
and
sa is ying
OEX
.
To any
such
X,
P
.
F
.
Smi h
e
all
associa e
h ee
classes
DX,
HX
and
EX
and
s udy
some
o
hei
closu e
p ope -
ies
unde
sui able
assump ions
on
X
.
This
no
only
led
hem
o
simple
p oo s
o
he
a o emen ioned
esul s o
Cha e s,
bu
also
o hei
module
heo e ic
gene aliza ions
.
Le
N,
G,
K
a
deno e
espec i ely
he
classes o
noe he ian
modules,
ini ely
gene a ed
modules
and
modules
o
K ull
dimension
G
a
.
The
module
heo-
e ic
gene aliza ions
ob ained
in
[4]
could
be
s a ed
as
ollows
.
(iii)
G
l
DN
=
N
(gene alizing
(i))
.
(i )
GnD
&
C
K,
:,
+
gene alizing
(ii))
.
These
a e
co olla ies 3
.3
and
2
.8
espec i ely
in
[4]
.
Sugges ed
by
"duali y"
in
he
ca ego y
R~mod
o
uni al
le
R-modules
we
associa e
o
X
h ee
mo e
classes
FX,
OX
and
FX
(see
Sec ion
1
o
hei
de ini ion)
.
The
s udy
o
some
o
he
closu e
p ope ies
o
hese
classes
leads
o
many
in e es ing
esul s
"dualizing"
he
esul s
o
P
.
F
.
Smi h,
Din
Van
Huynh
and
Nguyen
V
.
Dung
[4]
.
The
objec
o
he
p esen
pape
is
o
ca y
ou he
s udy
o
hese
closu e
p ope ies
and
p esen
p oo s
o
he
dual
esul s
.
Fo
ins an e
one
o
he
esul s
we
p o e
using
ou
me hods
is
he ollowing
:
( )
Le
M
be
a
semi-pe ec
module
in
he
sense
o
[13]
.
Assume
ha
ei he
M
is
ini ely
gene a ed
o
ha
M
is
ini ely
embedded
and
J(M)
is
small
in
M
.
Then
M
is
a inian
i
and
only
i
e e y
submodule
o
M
is
a di ec
sum
o
an
injec i e
moduleand
an
a inian
module
.
Ac ually
)
may
be
ega ded
as
wo
o ms
o
duals
o
(iii)
.
A
co olla y
o
)
is
he ollowing cha ac e iza ion
o
le
a inian
ings
.
( i)
A
ing
R
is
le
a inian
i
and
only
i i
is
semi-pe ec
and
e e y
le
ideal
o
R
is
a
di ec
sum
o
an
injec i e
le
ideal
and
an
a inian
le
ideal
.
1
.
The
classes
FX,
AX
and
I'X
We
will
be
wo king
in
he
ca ego y
R-mod
o
uni a y
le
R-modules
.
The
classes
X
o
R-modules
we
conside
will
always
be
assumed
o
sa is y
he ollowing
condi ions
a and b
.
a
.
ME_X
.
M'
-
M
=
:>
NI'cX
.
b
.
OEX
.
P oo
.
(i)
S aigh
o wa d
.
Oi
CERTAINCLASSES
oH
MODULES

1013
To
any
such
X,
P
.
F
.
Smi h
e
all
[4]
associa ed
h ee
clases
o
modules
( hough
hey
wo ked
in
he
ca ego y
mod-
.R,
o
igh
R-modules)
.
Be o e
ecalling
he
de ini ion
o
h ee
classes,
we
i s
explain
he
no a ion ha
we
will
be
adop ing
.
Fo
any
A/IcR,- nod,
we
w i e
N
<
11NI
o
indica e
ha
N
is
a
submodule
o
111
;
Né1VI
o
indica e ha
N
is
an
essen ial
submodule
o
111
-
and
N
«
AI o
deno e
ha
N
is
a
.
small
submodule
o
1VI
.
The
h ee
classes
DX,
TIX
and
EX
we e
de ined
as
ollows
in
[4]
.
DX
=
{AIcR-modIN
<
M
=111=K®L
wi h
N<K
and
K/N~!}
.
HX
=
{AIcR-modIN
<
M
=~>
II/NcX
}
EX
=
{AIcR-modIN
:Al
=
:>
AI/NEX}
.
Sugges ed
by
"duali y"
we
in oduce
he
ollowing
clases
:
FX=
{AIcR-modIN
<
M

M=K®L
wi h
K<N
and
N/KcX}
.
FX
=
{AIcR-modiN
<
AI

NcX}
OX
=
{McR-modIN
«
M

NcX}
.
As
in
[4]
when
he
ing
R
is
clea
om
he
con ex ,
M,
Z,
P,
I,
C, G,
N,
A,
U
;
K
will
deno e
espec i ely
he
classes o
all
R-modules,
he
ze o
modules,
p ojec i e
modules,
injec i e
modules,
semi-simple
mod-
ules,
ini ely
gene a ed
modules,
noe he ian
modules,
a inian
modules,
modules
o
ini e
uni o m
di nension
and
modules
wi h
K ull
di nension
<_
a
.
Recall
[11]
ha
116R-mod
is
said o
be
o
dual
Goldie
di nension
<_
k
i
h ee
exis s
no
su jec i e
map
M
_
W
>
N
l
x
...
xN,
.,
wi h
each
Ni
:~
0
and
>_
(k
+
1)
.
He e
k
is
an
in ege
>_
0
.
The
class
o
modules
o
dual
Goldie
di nension
<_
k
will
be
deno ed
by
H
k
.
We
w i e
S
o
he
class
cons i u ed
by
he
simple
modules
oge he
wi h
he
ze o
mod-
ule
.
We
will
nos ly
be
ollowing
he
no a ion
and
e minology
in
[4]
.
The
class
o
modules
o
ini e
dual
Goldie
di nension
(o
co ank)
will
be
deno ed
by
H
.
Lemma
1 .1
.
Le
X,
Y
be
classes
o
R-modules
(i)
I
X
C_
Y
hen
LX
C_
Y
whe e
L
s ands o
any
one
o
he
symbols
D,
H, E,
,
F
o
0
.
(ii)
FX
=
F(FX)
C
X
.
(iii)
C
C
FX
.
(i )
Fx
C
F(I
(D
X)
C
F(I
®
X)
=
F(X)
=
(X)
C
A(x)
.
( )
i n
x
c
F((D
x)
.
101
4

K
.
VARADARAJAN
(ii)
F om
he- e y
de ini ion
o
FX
i is
clea
ha
FX
C
X
.
Hence
(i)
abo e
yields
F(FX)
C
FX
.
Le
McFX
and
N
<_
M
.
Le
N'
<
N
.
Then
N'
<
M
;
hence
N'6X
yielding
NcFX
.
This
in
u n
implies
ha
AJEF(FX)
;
hence
FX
C
F(FX)
.
(iii)
Le
McC
and
N
<
M
.
Then
M
=
N
®
L
o
some
L<
M
.
Hence
he
choice
K
=
N
ul ills
he
equi emen
o
M
o be
in
FX
.
(i )
Since
X
C_
I
®
X,
om
(i)
we
ge
FX
C
F(I
®
X)
.
Le
McF(I
®
X)
and
N
<_
M
.

Since
McF(I
®
X)we
ge
NeI
®
X
.
Thus
M
=
0
®
M
and
N/0
-
NeI
®X
.
This
means
McF(I
®
X)
.
Hence
F(I
®
X)
C_
F(I
®
X)
.
Because
o
(i),
o
p o e
he
equali y
F(I
®
X)
=FX
we
ha e
only
o
show
ha
F(I
®_
X)
C
FX
.
Le
Mc (I
®X)
and
N
<
M
.
Then
M
=
K
®
L
wi h
K
<_
N
and
N/K6I
®
X
.
F om
K
<
N
we
ge
N
=
K
®
(L
nN)
;
hence
L
n
N
-N%K
eI
®X
.
This
yields
L
n
N
=A®B
wi h AeI,
BcX
.
Since
AeI and
A
_<
L
we
could
w i e
L
=
A
®C
wi h
CeM
.
Thus
M
=
K
®L
=
K
®A®C
.
Also
K®A<N
.
HenceN=K®A®
(CnN)
.
AlsoA_<LnN==>
L
n
N
=
A
®(C
n
N
n
L)
=
A
®(C
n
N)
since
C
<
L
.
F om
A®B=LnN=A®(CnN)
wege B-(LnN)/A-CnN
yielding
C
n
NeX
.
Also
M
=
K
®A
®
C
wi h
K
®A
<_
N
and
N/(K
®
A)
-C
n
NeX
.
This
p o es ha
Mcl'X
.
Hence
F(I
®
X)C
FX
.
To
comple e
he
p oo
o
i )
we
ha e
only
o
show
ha
X
C
AX
.
Le
M6F_X
and
N
«
M
.
Then
M
=
K
®
L
wi h
K
<
N
and
N/KEX
.
F om
K
<_
N«M
we
ge
K
K
M
.
Since
K
is
a
di ec
summand
o
M
his
implies
ha
K
=
0
;
hence
NeX
showing
ha
Mc0_X
.
( )
Le
MeI
n
FX
and
N
_<
M
.
F om
MEFX
we
ge
M
=
K
®
L
wi h
K
<_
N
and
N/KEX
.
Then
N
=
K
®
(L n
N)
yielding
N/K
-
Ln
NEX
.
Also
MeI
==>
KeI
;
hence
NEI
®
X
.
This
means
M6F(I
®
X)
yielding
I
n
FXC
F(I
®
X)
.
s
Be o e
s a ing u he
esul s
le
us
ecall
om
[41
he
de ini ion
o
SX
,
QX
and
PX
.
SX
=
{NIN
<M,
McX}
.
QX
=
{M/NIN
<
M,
M6X}
.
PX
=
{MI
he e
exis s
a
ini e
chain
0
=
No
<
N
l
<

<Nk
=
M
wi h
Ni/N2_leX
o
1
<
i
<
k}-
X
is
said
o
be
S
( esp
Q
o
P)
closed
i
SX
C_
X
( esp
.
QX
C
X
o
PX
C
X)
.
ON
CERTAINCLASSES
OF
MODULES

1015
Lemma
1 .2
.
Le
X
be a class
o
R-modules
.
Then
(i)
FX,
~X,
FX
a e
all
S-closed
.
(ii)
I X
is
S-closed,
hen
X
C_
FX
and
XC
C_
OX
.
(iii)
FX
®
X
=
FX
i
X
is
{S,
P}-closed
.
(i )
F(I
®
X)
=
(I
®
X)n
FX
i
X
is
{S,
P}-closed
.
( )
FX
is
Q-closed
i
X
is
Q-closed
.
P oo
..
(i)
Tha
F_X
is
S-closed
is
clea
.
Le
MEIX
and
M'
<
M
.
Le
N'«
M'
.
Then
N'
«M
and
hence
N'EX
.
This
means
M'EOX
.
Le
MEFX
and
M'
<_
Al
.
Le
N
<_
M'
.
F om
MEFX
we
ge
M=K®Lwi hK_<NandN/KEX
.
F om
K_<N<M'we
ge
M'=
K
®
(M'
n
L)
.
Clea ly
N/KEX
;
hence
M'cI'X
.
(ii)
Le
MEX
and
N
<_
M
.
Since
X
is
S-closed
we
ha e
NEX
.
Thus
M
=
0
®
M
wi h
N/0
-NEX,
yielding
MEFX
.
Hence
X
C
FX
.
Le
MEX
_C
.

Then
he e
exis s a
K
_<
M
wi h
KE_X
and
M/KEC
.
Le
N
«
M
.
Then
N
<_
J(M),
he
Jacobson
adi-
cal
o
M
.
I
l
:
M
-->
M/K
deno es
he
canonical
quo ien
map
we
ge
n(N)
<
97(J(M))
<
J(M/K)
=
0 Since
MlKEC
.
Hence
N
_<
K
.
Since
X
is
S-closed
we
ge
NeX
.
Thus
McAX
yielding
_XC
C_
21X
.
(iii)
Le
MEFX
®
X,
say
M
=
A®
B
wi h
AEI'X,
BEX
.
Le
N
<_
Al
.
Since
AEFX
we
ge
A=
K®L
wi h
K
<
NnA
and
(NnA)/KeX
.
Thus
M
=
K
(D
L
®B
and
M/A
-
BeX
.
The
exac ness
o
0
-,
N/(N
n A)
->
M/A
oge he
wi h
he
S-closed
na u e
o
X
yields
N/(N
nA)cX
.
The
exac ness
o
0
->
(N
n
A)/K
N/K
->
N/
(NnA)
->
0 and
he
P-closed
na u e
o
X
imply
ha
N/KEX
.
Hence
MEFX,
yielding
I'X
®
X
C
FX
.
The
e e se
inclusion
FX
C_
FX®
_X
is
ob ious
.
(i )
F om
lemma
1
.1(ii)
and
(i )
we
see
ha
F(1E)X)
C
(I®
X)n X
.
We
can
w i e
M
=
A
®
B
wi h
AEI,
BeX
.
F om
lemma
1
.2(i)
we
see
ha
AEFX
.
Le
N
_<
M
.
Since
AEFX
we
ge
A
=
K
®
L
wi h
K
_<
A
n
N
and
An
N/KEX
.
Hence
M
=
K
®
L
®B
.
F om
K
<_
N
we
ge
N
=
K
®
(L
®
B)
n
N
.
Also
AEI
=>
KeI
.
The
exac ness
o
0
->
N/(N
n A)
->
M/A
and
0
->
(A
n
N)/K
-~
N/K
->
N/
(A n
N)
-~ 0
and
{S,
P}-closedness
o
X
immedia ely
yield
N/KEX
.
Bu
N/K-Nn(L®B)
.
Hence
NEI®X,
p o ing
ha
MEF(I
®
X)
.
Hence
(I
(D
X)n
FX
C_
F(I
®
X)
.
( )
Le
MEFX
and
N
<
M
.
Any
submodule
o
M/N
is
o
he
o m
L/N
wi h
N
_<
L
_<
M
.
F om
MEFX
we
in e
LEX
.
Since
X
is
Q-closed
we
ge
LINEX
.
This
implies
ha
M/NEFX
.
Rema ks
1
.3
.
Lemma
1
.1( )
in
[4]
also asse s
ha
EX
is
S-closed

101
6

K
.
VARADARAJAN
i
_X
is
S-closed
.

The
dual
esul
i
i
we e
ue
would, be
,
ha
AX
is
Q-closed
whene e
X
is
Q
closed
.
We
now
.
gi e
an easy
esample
o
show
ha
he
dual
esul
is
no
ue
.
Le
Z
deno e he
class
consis ing
o
he
ze o
modules
in
Z-mod
.
Clea ly
Z
is
Q-closed
.
Also
AZ
= {M6Z-mod
jJ(M)
=
0}
.
Clea ly
Z6OZ,
bu
Z
p
2
1
OZ
o
any p ime p
.
This
shows
ha
áZ
is
no
Q-closed
.
P oposi ion
1 .4
.
Le
X
be
any
{S,
P}-closed
amily o
modules
.
Then
FX
=FX®
X
®
(P
nFX)
.
P oo
.
We
need
only
p o e
he
inclusion
FX
®
X
®
(P
nFX)C
FX
.
F om
lemma
1
.2(i )
we
ha e 1'X
®
X
=
FX
.
Hence
i
su ices
o
p o e
ha
FX
®
(P
n
FX)
C_
FX
.
Le
M
=
A®B
wi h
Ac X
and
BeP
n
FX
.
Le
N
<_
M
and
PB
M
=A®
B
-->
B
he p ojec ion
on o
B
.
F om
Be X
we
ge
B
.
=B
l
®
B2
wi h
Bl
<
PB(N)
and
PB(N)/BlEX
.
F om
BcP
we
ge
B
1 E
P
and
B2EP
.
Le
n
=PBINn(A®B1)
:
Nn(A®B
1 )
-~
B,
.

Since
B
l
<PB
(N)
we
see
ha
a
:
N
n
(A
®BO
->
B
lis
on o
.
Since
B
1
6P,
he e
exis s a spli ing s
:
B
l
->
N
n
(A
®
Bl)
o
a
.
Le
N'
=
s(Bl)
.
Then
N
n
(A
®
Bl)
=
N
I
®'Ke
a=
N'
-
®
(Nn
A)
.
F om
AcFX
we'
ge
A
=
A
1
®
A2
wi h
A,'<_
N
n
A
and
(N
n
A)/A
l
eX
.
Again,
NnA
=
Al
®(NnAnA
2 )
=
A1®(NnA2)
yields
NnA
2
-
(NnA)/AleX
.
Conside ,
pB/A
®B
l
:
A®
Bl

+
B,
.
Clea ly
s
is
also
a
spli ing
o
pB/A
®
B,
.
Since
Ke
pB/A
®B
l
=A
we
see
ha
A®
N'
is
-
ano he
in e nal
di ec
sum
ep esen a ion
o
A®
B,
.
Hence
M
=
A®B=
A®Bl®B2=A®N'®B2=A1®
A2
.®N'®
B
2
.
Since
Al
®N'<N
wegé N=A
l
®N'eNn(A2TB2)
.
.Le y=PBINn(A2®B2)
:
N
n
(A2
®
B2)
-
B2
.
Since
pB(Al
®
N')
<
B
l
and
B=
B
l
®
B2
we
see
ha
PB(N)
nB2
=
PB((A2
®
B2)
n
N)
=
Image
y
.
Bu
PB(N)
_
B
l
®
(PB(N)
n
B2)
;
hence
Image
y
=
pB(N)
n
B2
-
PB(N)/B,
is in
X
.
Aslo
Ke
y
=
N
n
A
2
EX
.
Since
X
is
P-closed
we
ge
N
n(A2
®
B2)6X
.
Also
N%(A1
®
N')
-
N
n(A
2
(D'B2)cX
.
This
shows
ha
MEFX
.
Thus
FX®
X
®
(P n
FX)
CFX
.
This
comple es he
p oó
o
p oposi ion
1
.4
.
E
Lemma
1 .5
.
I
X
is
Q-closed
hen
OX
is
closed u ide
minimal
epi-
mo phic
images
.
P oo
.
Le
McAX
and
M
-~
M"
aminimal
epimo phism
(Le
.
Ke
e
«
M)
.
Then
N"
«
M"

c
-1
(N")
«
M
.
In
pa icula
N"
«
M"
=~> e
-1
(N")
«
M
=>
e
-1
(N")cX

N"eX
(since
X
is
Q-closed)
.
This
p o es
ha
M"6AX
.
Be o e
p oceeding
u he
we
need
o
ecall
some
de ini ions
and
e-
sul s
om
[7],
[111,J12]
;
[13]
.
Le
N
<
M
.
Then
K
<
M
is
called
a
supplemen
o
N
in
M
i
ON
CEI TAIN
CLASSES
O
MODULES

1017
(a)
K+N=hand
(b)
K'<K,K'+N=A,1=~>
K'=K
.
I
is
known
ha
K
is
a
supplemen
o
N
in
M
i
and
only
i
K+N
=
M
and
K 1N
«
K
(Le nma
6
.2
in
[13])
.
In
[13]
we
called
a
module
M
semi-
pe ec
i
o
e e y
N
<
AJ
he e
exis s
a
supplemen
in
M
(De ini ion
6
.6
in
[13])
.
In
[11]
we
e e ed
o
his
as
p ope y
(P
l )
o
M
.
The
module
M
is
said
o
ha e
p ope y
(P2)
i
o
any
L
_< Al1,
N
<_
M
sa is ying
L
+
N
=
M
he e
exis s
a
supplemen
K
o
N
in
AJ
sa is ying
K
_<
L
.
I
M
has
p ope y
(Pi)
hen
any
quo ien
module
o
M
has
p ope y
(P
i
)
o
i
=
l,
2
(P oposi ion
6 .20
in
[13]
and
P oposi ion
2
.29
in
[11])
.
Clea ly
P
2
=>
PI
.
Lemma
1.6
.
Le
X
be
Q-closed
and
AJEAX
.
Assume
u he
ha
AJ
has
p ope y
(Pi)
.
Then
e e y
epimo phic
image
o
M
is
in
OX
.
P oo
..
Le
q
:
M
->
AJ" be
any
epi no phism
and
N
=
Ke
91
.
Le
K
be a
supplemen
o
N
in
M
.
Then
K
+
N
=
AJ
and
K
1
N«
K
.
In
pa icula
71/K
:
K
->
M"
is
a minimal
epi no phism
.
F om
lemma
1
.2(i)
we
ge
KE,~,X
.
Now
lemma
1.5
yields
M"eáX
.
Example
1 .7
.
(a)
Le
T
deno e
he
class
o
o sion
abelian
g oups
.
In
Z-mod,
T
is
{S, P,
Q}-closed
.
In
[4]
he
class
DT
is
comple ely
de e mined
(P oposi ion
1 .6
o
[4])
.
I is
easy
e
see
ha
ET
=
A%1
and
ha
HT
=T=
FT
.
Fo
any
AJcZ-mod
le
J(A4) deno e
i s
Jacobson
adical
.
Since
J(A11)
is
he
sum
o
all
s nall
submodules
o
All
we
see
inmmedia ely
ha
AT=
{MEZ
-mod
jJ(A11)cT}
.
F om
lemma
1
.2(i)
we
know
ha
FT
is
S-closed
.
Since
he
only
di ec
su nmands
o
Z
a e
0 and
Z
i
ollows
ha
Z
1
F
T
.
Combining
his
wi h
he
S-closed
na u e
o
F
T
we
see
ha
FT
C_
T
.
Also
lem na
1
.2(ii)
implies
T
C
FT
.
Hence
FT
=T
.
(b)
Le
T'
deno e he
class
o
o sion
ee
abelian
g oups
.
Then
T'
is
S-closed
.
I
is
i ial
o see
ha
FT'
=
T'
.
Suppose
A FT'
.
Since
he
only
o sion ee
ac o
g oup
o a
o sion
abelian
g oup
is
0
we
see
ha
any
N
<
(AJ)
is
a di ec
summand
o
1Vl
(he e
(M)
deno es
i e
o sion
subg oup
o
M)
.
I
ollows
ha
any
N
<_
(M)
is
a
di ec
summand
o
(M)
and
ha
(M)
i sel
is
a
di ec
summand
o
M
.
Thus
(M)cC
and AJ
=
(M)
®
L
wi h
LcT'
.
This
yields
FT'
C_
C®
T'
.
Also
AcC
<~-->
A
=
(A)
and
p
(A)
is
a
ec o
space
o e
Z
p
o
e e y
p ime
p
.
Le
M
=
A®
B
wi h
AcC
and
BET'
.
Le
N
<AL
Then
(N) <
(1V1)
=
A
.
Since
AEC
we
ge
A =
(N)
®
L
and
101
8

K
.
VARADARAJAN
bo h (N) and
L
will
be
in
C
.
F om
M
= A®
B=
(N)
®
L
®B
and
N/ (N)cT_'
we
see
ha
McI'T'
.
Hence
C
®T'
C_
I'T'
.
Using
he
e e se
inclusion
al eady
p o ed
we
ge
FT'
=
CT
T'
.
F om
lemma
1
.1(i )
we
ha e FT'
C
OT'
.
We
will
ac ually
gi e
a
com-
ple e
cha ac e iza ion
o
he
class
áT'
o n
which
i
will
ollow
i n nedi-
a ely
ha
he
inclusion
FT'
C_
áT'
is
a
s ic
inclusion
.
Le
M6AT
'
.
Suppose
o
some
p ime
p,
he
p-p ima y
o sion
p
(M)
o
M
is
non-ze o
.
Then
he e
exis s
a copy
o
Z
p
in
p
(M)
.
Suppose
N
<_
M
sa is ies
Z
p
+
N
=
M
.
Ei he
N
1
Z
p
=
Z
p o
N
n
Z
p
=
0,
in
he
o me
case
N
=
M
and
in
he
la e
case
M
=
N
®
Z
p
.
I
o
all
N
<_
M
sa is ying
Z
p
+
N
=
M
we
ha e
N
=
M,
hen
Z
p
«
M
and
his
con adices
he
assump ion
ha
McAT'
.
Hence
M
=
Z
p
®
N
o
some
N
<_
M
.
Thuswe
ha e
shown
ha
i
p
(M)
:7É
0,
any
copy
o
Z
p
in
p
(M)
is
a
di ec
summand
o
M
.
In
pa icula
his
implies
ha
he e
a e
no
elemen s
o
o de
p
2
in
p
(M),
hence
p
(M)
is
a
ec o
space
o e
Z
p
.
Hence
(M)
=®
p
p
(M)
is
in
C
.
We
claim
ha
(4)

n
T'
=
{MeZ-mod
/
any
Z
p
<
M
o
any
p ime
p
is
a di ec
summand
o
M}
.
Because
o
he
obse a ions
in
he
ea lie
pa ag aph,
o
p o e
(4)
we
ha e
only
o
show
ha
i
MeZ-mod
has
he
p ope y
men ioned
in
he
igh
hand
side o
(4)
and
i
N
«
M
hen
NcT'
.
I
on
he
con a y
he e
is
an
N
«
M
wi h
N
0
T',
hen
p
(N)
,-É
0
o
some
p ime
p
.
Then
he e
is
a
copy
o
Z
p in
p
(N)
.
Since
N
«
M
i
will
ollow
ha
his
copy
o
Z
p is
small
in
M
.
Howe e ,
any
Z
p
<
M
being
a di ec
summand
o
M
canno
be
small
in
M
.
F om
(4)
we
see
ha
(di ec
p oduc
o e
all
p imos)
is
in
AT'
.
Howe e ,
(M)
=®
p
Z
p
and
i is
well-known
ha
(M)
does
no
spli
o
o n
M
.
Hence
111
«
FT'
.
This
p o es
ha
he
inclusion
FT'
C
áT'
is
s ic
.
2
.
S udy
o
AX
when
X
=
A
n
H,
Fo
esul s
on
dual Goldie
dimension
o
co ank
he
eade
may
e e
o
[7],
[111
.
As
al eady
ema ked
in
[11],
i
he
dual Goldie
dimension
ON
CER
.TAIN
CLASSES
OF
MODULES

101
9
o
M
is
in ini o
we
canno
asse
ha
he e
exis s
a
su jéc i e
map
cp
M
->
11'
1
Ni
wi h each
Ni
:,A
0
.
(See
P oposi ion
1.6
in
[11])
.
All
we
can
asse in
his
case
is
ha ,
gi en
any
in ege
d
>_ 1
we
can
ind
a
ce ain
su jec ion
9
:
M
-->
Il,q-
1
Lj
wi h
each Lj
7~ 0
( he
modules
L
.-
in
gene al
will
depend on
d
)
.
This
di e en
beha iou
o
dual
Goldie
dimension
as
compa ed
o
Goldie
dimension
necessi a es
many
changos
in
he
o mula ion
and
in
he p oo s
o
esul s
dual
o
hose ob ained
in
Sec ion
2
o
[4]
whe e
he
heo y
o
Goldie
dimension
plays
a
c ucial
ole
.
We
i s
obse e
ha
he
class
H
kis
Q-closed
.
Lemma
2
.1
.
Le
X
be
Q-closed
wi h
X
C_
H
k
.
Le
AllcOX
and
N
_<
M
sa is y
N
+
J(A11)
=
AJ
.
Assione
ha
Al
has
p ope y
(P1)
.
Then
M/NcH
k
.
P oo
.,
Le
l
:
Al
->
M/N
deno e
he
quo ien
map
.
F om
N
+
J(M)
=
M
we
ge
l(J(M))
=
M/N
.
Hence
J(M/N)=
M/N
.
Suppose
i
possible
ha
M/N
has
dual
Goldie
dimension
>
k
.
Then
he e
exis s
a
su jec ion
cp
:
M/N
--->
A1 x
. . .
x Ae
wi h
2
>
k and
each
A
j
:y~
0
.
F om
J(M/N)
=
M/N
we
ge
J(A
;)
=
Aj
o
1<_
j
<_
Q
.
Since
J(Aj)
=
Aj
,-á
0
and
J(A
;)
is
he
sum
o
all
small
submodules
o
A
j
we
see
ha
he e
exis s
a Bj
«
Aj
wi h
B
. :~
0
.
Then
B
1
x
. . .
x Bg
«
A
1
x
. . .
x
Ae
.
F om
lemma
1
.6
we
ge
A1
x
.
.
.
x
A
e
cOX
.
This
implies
B
1
x
.
. .
x
BQcX
.
This
con adic s
he
assump ion
ha
X
C_
H
k
,
since
co ank
B
1
x
. .
.
x
BQ
>
2
>
k
.
Co olla y
2
.2
.
Suppose
X
is
Q-closed
and
_X
C_
H
k
.
Le
McáX
.
Suppose
M
Izas
p ope y
(P1)
and,
sa i
.s ies,I(M)
=
M
.
Then
McH
k
.
P oo
::
Choose
N
=
0
in
lemma
2 .1
.
P oposi ion
2
.3
.
Le
X
be
Q-closed
wi h
X
C_
H
k
.
Le
Mci1X
and
assume
ha
M
has
p ope y
(P1)
.
Then
he e
exis s
an
N6X
such
ha
114'/N
=
B
e
H
wi h
B¿C
and
HEH
k
n
OX
.
P oo
.:
Le
L
be
a
supplemen
o
.
.I(A4)
in
AJ
.
Then
L+
J(M)
=
M
and
LnJ(M)
«
L
.
Also
L/(L
n
J(M))
-
AJ/J(AJ)e0X
by
lemlna
1 .6
.
Since
OX
is
S-closed
(lemma
1
.2(i))
we
ge
LcOX
.
F om
LnJ(M)
«
L
we
ge
Ln
J(M)cX
.
Since
M/J(M)
has
p ope y
(P1)
(P oposi ion
6
.1
in
[13])
and
J(M/J(M))
=
0
om
p oposi ion
3
.3
in
[11]
we
see
ha
M/J(M)cC
.
Hence
L/(LnJ(M))EC
.
F om
lemma
2
.1
we
ge
MILc_H
k
.
l
we
se
N
=
LnJ(M)
we
ge
NeX
and
0
->
L/N
->
M/N
->
MIL
->
0
exac
.
102
6

K
.
VARADARAJAN
We
will
now
cha ac e ize he
class
V(T')
.
We
will
show
ha
V(!
:')
=
{McTI
J(M)
=
0}
.
Le
McV(T')
.
We
will
show
ha
0
is
he
only
small
submodule
o
M
.
Then
i
ollows
ha
J(M)
=
0
.
Suppose on
he
con a y
0
:,A
N
«
M
.
Since
V(T')
C_
T'
we
ha e
MeT'
.
Hence
NcT'
.
This
means
he e
is
a
copy
o
Z
in
N
.
Conside
he
subg oup
2Z
o
Z
.
F om2Z
<_
Z
<_
N
«
111
we
ge
2Z
«
M
.
Now,
M/2Z
has
non-ze o
2 o sion,
con adic ing
he
ac
ha
McV(T')
.
Con e sely
any
MeT'
wi h
J(M)
=
0
is
clea ly in
V(T')
because
hen
0
is
he
only
small
submodule
o
M
and
M/0
-
MeT'
.
This
p o es
(5)
.
F om
(5)
we
see
ha
he
inclusion
V(!
:')
C
T'
is
a
s ic
inclusion
;
because
Q6T'
bu
Q
«
V(T')
since
J(Q)
=
Q
.
We
included in o ma ion
en
he
classes
LT,
VT,
LT'
and
VT'
o
comple e
he
examples
discussed
in
1
.7
.
Re e ences
1
.

A
.
W
.
CHATTERS
;
A
cha ac e iza ion
o igh
Noe he ian
ings,
Qua e ly
J
.
Ma h
.
Ox o d
33
(1982),
65-69
.
2
.

DINH
VAN
HUYNH
AND
PHAN
DAN,
On
ings
wi h
es ic ed
min-
imum
condi ion,
A ch
.
Ma h
.
51
(1988),
313-326
.
3
.

DINH
VAN
HUYNH
;
NGUYEN
V
.
DUNG
AND
PATRICK
F
.
SMITH,
Rings
cha ac e ized
by
he igh ideals
o
cyclic
modules,
P oceed-
ings
o
he
Edinbu gh
Ma hema ical
Socie y
32
(1989),
355-362
.
4
.

PATRICK
F
.
SMITH,
DIN
VAN
HUYNH
AND
NGUYEN
V
.
DUNG,
A
cha ac e iza ion
o
Noe he ian
modules,
Qua
.
J
.
Ma h
.
Ox o d
41
(1990),225-235
.
5
.

DIN
VAN
HUYNH,
NGUYEN
V
.
DUNG,
AND
PATRICK
F
.
SMITH,
A
cha ac e iza ion
o
ings
wi h
K ull
dimension,
J
.
Alg
.
132
(1990),
104-112
.
6
.

DINH
VAN
HUYNH
AND
NGUYEN
V
.
DUNG,
A
cha ac e iza ion
o
a inian
ings,
Clasgow
Ma h
.
J
.
30
(1988),
67-73
.
7
.
B
.
SARATH
AND
K
.
VARADARAJAN,
Dual
Goldie
dimension
II,
Communica ions
in
Alg
.
7
(1979),
1885-1899
.
8
.

P
.
F
.
SMITH,
Some
ings
which
a e
cha ac e ized
by
ini ely
gene -
a ed
modules
;
Qua
.
J
.
Ma h
.
Ox o d
29
(1978),
101-109
.
9
.

P
.
F
.
SMITH,
Rings
cha ac e ized
by
he
cyclic
modules,
Canadian
J
.
Ma h
.
30
(1978),
98-111
.
10
.

P
.
VAMOS,
The
dual
o
he
no ion
o
ini ely
gene a ed,
J
.
London
Ma h
.
Soc
.
43
(1969)
.

ON
CERTAIN
CLASSES
OF
MODULES

1027
11
.
K
.
VARADARAJAN,
Dual
Goldie
dilnension,
Communica ions
in
Alg
.
7
(1979),
565-610
.
12
.
K
.
VARADARAJAN,
Modules
wi h
supplemen s,
Pac
.
J
.
o
Ma h
.
82
(1979),559-564
.
13
.
K
.
VARADARAJAN
AND
P
.
R
.
WANI,
Modules
o e
endomo phism
ings
11,
Ac a
llla h
.
HungaHca
53
(1989),
309-337
.
14
.
K
.
VARADARAJAN,
Hop ian
and
co-Hop ian
objec s
( o
appea )
.
Depa men
o
Ma hema ics
and
S a is ics
The
Uni e si y
o
Calga y
2500
Uni e si y
D i e
N
.W
.
Calga y,
Albe a
LANADA
T2N
1N4
P ime a
e sió
ebuda
el
15
de
No emb e
de
1991,
da e a
e sió
ebuda
el
2de
Ma 4
de
1992