Pub
.
Ma
.
UAB
Vol
.
30
n2
2-3
Des
.
1986
LINEARY COMPACT INJECTIVE MODULES
AND
A
THEOREM
OF
VAMOS
Fo Ka y
and
Pe e
Ca l
Fai h
1
A
ing
R will
deno e
a
commu a i e associa i e
ing
wi h uni
.
A e
Vámos,
R
is
a
SISI
Ring
i
e e y
subdi ec ly
i educible ac o
ing
is
sel -injec i e
.
Le
M
be
a
maximal
ideal
o
R
and
le E =
E(R/`4)R
deno e he injec i e
hull
o he simple'R-module
R/M,
and le
A(M)
deno e he
endomo phism
ing
.
Now
E
is
canonically
a
module
o e
he
local
ing
RM o
R
a
M,
and he
unique
simple
RM
-module
embeds
in
E
canonically
.
Mo eo e
:
E =
E
(
R/M
)R
=
E(R i/MRCR)RM
We
call
he
module
(A)
he local
injec i e
hull
o
R a
M,
and i s endomo phism
ing
A(M)

=End
E(R/M)
R = End
E
R-

(B)
he-local
endomo phism
ing
o
R
a
M
.
1
A
pa
o
his
pape
was
w i en
sp ing semes e
1986
a he
CRM
o
Ins i u
d'Es udis
Ca alans
o
Ba celona,
while
I
was
holding
a
Ru ge s
Uni e si y
Facul y
Academic
S udy
P og am
(FASP)
.
I
ha e
he
pleasu e
o
hanking
P o esso
Pe e
Menal
o
,
in i ing
me
o
collabo a e,
P o esso
Ja mie
Moncasi
o
his
many
co dial
a angemen s
on my behal ,
and
o he
membe s
o
he
Facul y
and
s a
o
helping
o
make
he
s ay
such
happy
and
ma hema ically p o i able
one
.
127
R
is
Vamosi
an
(classical
in [1]) i
e e y
local
injec-
i e hull
o
R
is
linea ly
compac
(in
he
disc e e
opology)
.
In
[1]
Vamosp o ed
ha
e e y
Vamosian
ing
is
SISI,
and
ha e e ylocalendomo phism
ing
o
a
SISI ing
is
commu a i-
e
.
We
shall
p o e
he
con e se
he e,
and
numbe
o
subsidia y
esul s
:
1
.
Any
SISI
chain ing
is
Vamosian,
in
ac ,
an
almos
maximal alua ion
ing (Theo em
9)
.
As
a
consequence,
we p o e
:
2
.
A
ing
R
ha
is
locally
a
SISI
chain
ing
is
Vamos
ian
(Theo em
10)
.
Von
Neumann
egula ings
a e
locally
Noe he ian
ings,
and
a e
examples
o
Vamosian
ings
([l])
;
we
show ha
polyno-
mial ingso e
hem
a e
also
Vamosian
(Theo em
12
and
Co olla-
y)
.
A
numbe o
unsol ed
p oblems
a e
lis ed
.
One
o
he
main
ones asks
i
R
Vámos
(SISI)
is
inhe i edby
he
polynom-
i
.a
l
ing R[x]
.
This
is
unknown
e en
o
an
almos maximal al-
ua ion
ing
R
.
1
.
THEOREM
.
The
ollowing
a e
equi alen condi ionson
a
ing
R
:
(1)
R
is
SISI
.
(2)
E e y
local
endomo phism
ing
o
R
is
commu a i e
.
(3)
E e y
R-submodule
o
e e y
local
injec i e
hull
o
R
is
quasi-injec i e
.
(4)
E e y
R-submodule
o
e e y
local
injec i e
module
E
is an
End
R
E-submodule,
i
.e
.
is
ullyin a ian
.
When
any
o
hesehold,
hen
RM
is
SISI
o
e e ymaximal
ideal
M
.
Rema k
:
When
his
is
so,
hen
e e ylocal
endomo phism
ing
o
R
is
"almos "SISI
;
See
P oposi ion
4
.
Fo
he
p oo ,
we need
a
esul
implici
in
[1]
.
2
.
PROPOSITION
.
Le
E
be an
indecomposableinjec i e
R-module
.

'
The
ollowing
a e
equi alen
condi ions
.
(1)
E e y
submodule
-
o
E
is
quasi-injec i e
.
(2)
E e ysubmodule
o
E
is
ully
in a ian
(FI)
.
(3)
E e ycyclicsubmodule
o
E
is
quasi-injec i e
.
(4)
E e ycyclicsubmodule
o
E
is
FI
.
(5)
I
a
cyclicmodule
R/I
embeds
in
E
hen
R/I
is
a
sel -injec i e
ing
.
(6)

Fo
each
a
EA
=
End
ER
and
x E
E,

he e
exis s
E R
such ha
a(x)
=
x
.
When
his is
so,
hen
A =
End
RE
is
commu a i e
.
PROOF
.

(1)
<
*
-
(2)

by
a
heo emo Johnson
and
Wong
([7],
p
.63
.
Co
19
.3),
which
s a es
ha
an
R-module
M
is
quasi-injec i e
i
M
is
ully
in a ian
in
E(M
R
) .
I
e
e y
cyclicsubmodule
o

E

is
ully-in a ian
(quasi-injec i e)
hen
e e y
submodule
is,
hence
(1) -
(4)
a e
equi alen
.
Ob iously
(4)
-
(6)
.
Fu he mo e
(5)
-
(3),
because
R/I
sel -injec i e
implies
R/I
is
quasi-injec i e
(since
e -
e y
R-submodule
is an
R/I-submodule)
.
(3)
=»(5)
.
I
R/I
is
quasi-injec i e
qua
R-module,
i
is
quasi-injec i e
qua
R/I-module,
equi alen ly,
sel
-
injec i e

by
Bae 's
c i e ion

([6],
p
.157,
Theo em
3
.41
.)
E iden ly,
(6)
implies ha
A
is
commu a i e
.
3
.
COROLLARY
.
(Vámos)
A
ing
R
is
SISI
i
e e y
local
injec i e
module

E =
E(R/M)
R

sa is ies
any
o
he
equi alen
condi ionso
he
p oposi ion
.
PROOF
.
This
ollows
since
e e ysubdi ec ly
i educible
ac o
ing

R/I

embeds
in

E(R/M)
R
,

whe e

R/M
=
socle
R/I
;
and
con e sely,
i

R/I
-
E(R/M),
whe e

M

is
maximal,
hen
R/I
is
subdi ec ly
i educible
.
(1)
°
(2)
by Vámos
[1],
and
(1)
«
(3)
by
Co olla y
3
.
Mo eo e
in
iew
o
(A)
and
(B),
Co olla y
3
also
yields
(3)
-
(4)
.
(2)

(1)
.
I
su ices
o
p o e
ha
R
M
is
SISI
o
e e y
maximal
ideal
M,
hencesuppose
R
is
local
wi h
maximal
ideal

M
.

In his
case

E
=
E(R/M)
R

is
an

injec i e
cogene a o
o
mod-R
.
Nowle
M =
R/I
be
a
subdi ec ly
i educible
ac o
ing,
andle
E =
E(M
R
)
be
i s
injec i e
hull
aken
in
E
.
PROOF
OF
THEOREM
1
.
This
can
be donebecause
E
is
alsoessen ialo e
M
as an
R-module,
and

E =
E(
;_1
R
),
since
M
--4
-
E
and
E
is
indecom-
posable
.
Fu he mo e
M
=
ann
E
I
D
E,
and
M
is
essen ial
o e
M
as an
R,
whence
as
an
R-module,
so he e o e
M
=
E
.
This
shows
ha
E
is a
ully
in a ian
submodule
o
E
since
ob iously
whe e
K =
annAE
.
so
ha
aM
CM
V
aE
A
.

Thus
The
ac
ha
M
--->
E
implies
ha
E
is
a
cyclic
P
.-module
(e
.g
.,
see
[8],
p
.15,
P op
.
5
.5),
whence E
^A
in
mod-A
.-
Fu he mo e,
Q
=
Qmax(R)C-
A
A
=
A/K
^
End
ER
E
^A^
Biend
El
= Q

(C)
canonically
(e .g
.
[7],
p
.81,
P op
.
19
.21
.)
.
Since ER
is
injec i e,
hen
i
is
quasi-injec i e
o e
Q
=
Biend ER
by
Co olla y
5
.6A,
p
.15
o
[8)
.
Using
(C)
we
see
ha
A
is
sel -injec i e
.
We
also
need
he
ac ha
A
is
an injec i e
cogene a o
o e
R
(since
E
is)
.
Thus
e
:
e y
ideal

H

o

R

is
he
annihila o
o an
.A-submodule
o
E

A
,
.
say
H =
annIG
o
an
ideal
G
o
A
.
(See
[7],
p
.
190,
Co olla y
23
.23
.)
Bu
i
H
is
chosen
o
be
a
dense
ideal
o
R
(=
R
is
a
a ionalex ensiono

H),

hen
he
only
ideal
in
A
=
Qmax(R)

ha
annihila es
i
is
ze o
.
See,
e .g
.
[71,
p
.80,
19
.32(b),
which
implies ha
ann-annAH
= E
o
any
denseideal
H
o
R
.
In
ou
con ex ,
his
means
ha
ann
A
H
=
0,
so
G
=
0
whence
H
=
R
.
Bu ,
hen
R=
A,
since
o
e e y
q
E
Q
he
exis s
a
dense
ideal
I
o
R
wi h qI
--~
R
.
This
p o es
ha
R
is
sel -injec i e,
and
hence
R
is
SISI
.
Rema k
:
(1)
A
p oo
o
(2)
-
(1)
has
kindlybeenp o idedby
P o esso
Vámos,
who
also
supplied
he
ollowingexample
(2)
o
a
locallyVamosian
ing
ha
is
no
SISI
.
Suppose
(2)
holds
bu
(1)
ails
.
As
be o e,
we
may
assume
ha
R
is
a
local
subdi ec ly
i educible
ing
embedded
in
E
=
E(R)
.
Since
E
>,¿
R
he eexis s
xE
E R,
and
since
E
is an
injec i e
cogene a o ,
he e
exis
homomo phisms
such ha
Pa iallyDi e en
P oo
E
-->
E
and
R
:
E
--~
E
a
(1)

= x,

Q
(1)

=
0,

0(x)

y,
0
.
Then
0~ G (x)

=

a
(1)

76
Cío
(1)

=
0
con adic ing
commu a i i y
o
End
RE
.
(2)
Le
R
be
a
subdi ec ly
i educiblealmos
maximal
o ch
ing,
i
.e
.
a
ing
R
wi h
:
(i)
a
leas
wo
maximal
ideals
such ha
R
M
is
an almos maximal alua ion ing
o
each
ME
max
R,
(ii)
a
wais
P,
whe e
P
is
a
minimalp ime
and
a
unise ial
module,
and
such ha
R/P
is
an h-local
do-
main
[i
.e
.
e e y
nonze o
p ime
is
con ainedin
a
unique
maximal
ideal,
and
e e ynonze oidealo
R/P
is
con ained
in
jus
ini ely
manymaximal
ideals
.]
Such
ingsexis
(see
[15])
bu
canno
be
SISI
since
R
is
no
sel -injec i e
.
(An
indecompos-
ablesel -injec i e ing
is
local
.)
The
nex
esul
shows
ha
a
localendomo phism
ing
o
a
SISI ing
is
"almos "
SISI
.
4
.
PROPOSITION
.
I
E
is
an
injec i e
R-modulewi h
commu a i e
endomo phism
ing
A,
any
A-submodule
o
E
is
quasi-injec i e,
and
A
modulo
any
ideal
I
such ha
A/I
C-
.
E

is
sel -injec i e
;
equi alen ly

A/I

is sel
injec -
i e o any
ideal
I
=
ann
A
x
o
some
x E E
.
PROo
.
As
s a ed
in
he
p oo o Theo em
1,
E
A
is
quasi-injec i e,and
A
=
End
AE
.

I
S C M
a e
A-submodules
o
E,
and
:
S +
M
an
A-map,
hen
by
quasi-injec i i y
o
E
o e
A, is
induced
by
a
EA
.
Since
M
is
an
A-submodule,

a
MC
M,

hence
ex ends
o an
endomo phism
o
.
M
A
.
This
p o es
quasi-injec i i y
o
any
A-submodule
M
o
E
.
The
sel -injec i i y
o
A/I
ollows om'i s
quasi-injec-
i i y
as in
he
p oo o
P oposi ion
2
.
I
:
A/I
->
E
is
an
embedding
o A-modules,
hen
I
=
ann
A
x,
whe e
x =
(1+I)
.
Con e sely,
i
I =
ann
Ax
hen
he e
is an
embedding
A/I
---->
E
sending

A +
I
-
ax
G
a
EA
.
No e,
i
R
is
SISI,
hen
e e y
local
endomo phism
ing,
A
=
End
E(R/M)
R
,
is
commu a i e,
and
he
uniquesimple
A-mod-
ule
W
embeds
in
E
and
coincideswi h
V
.=
R/M
.
Thus,
he
p oposi ion
would
imply ha
A
is
SISI
p o idedonly
ha
E
is
injec i eo e
A
.
This
is
no
in
gene al
ue
o
a
SISI
ing
R
.
In
ac ,
Vámossingles
ou
a
classo ings
(called
classical
in [1J) o
ec i y
his
de iciency
.
We
say
ha
.
R
is
a
Vámos
ing
,
o Vamosian
( o me ly
classical)
p o ided
ha
e e ylocalinjec i e
hull
is
linea ly
compac
(l .c
.)
o e
R
.
We employ
he
e minology
injec i endo
o
indica e
when
a
module
F
o e
R
is
injec i eo e
i s
endomo phism
ing
A
.
An ideal
I
is
co-subdi ec ly
i educi-
ble
(co-SDI) i
R/I
is
a
subdi ec ly
i educible
ing
.
VÁMOS
THEOREM
[1]
.
I
R
is
Vamosian
hen
:
(V1)
R
is
SISI
.
(V2)
The
local
endomo phism
ing
A
a
any
maximal
ideal
M
is
he
.comple ion
o
R
M
in
he
opology
gene a ed
by
he
co-SDI ideals
o
R
M
,
and
is
a
l .c
.
ing
.
(V3)
E e ylocalinjec i e
hull
E
is
injec i endo,
and
l
.c
.
o e
i s
endomo phism
ing
A,
equi a-
len ly
Hom
A
(
E)
induces
a
Mo i a
duali y
in
mod-A
(on
he
ull
subca ego y
o
l .c
.
A-modules)
.
(V3)
Follows
om
heo ems
o
Mo i a
[4)

and
Muelle
[3],
which
imply ha
a
commu a i e
ing
A
has
a
Mo i a
dua-
li y
i he
leas
injec i e
cogene a o
E
o e
A
sa is ies
A
=
End
A
E
.
By
Muelle
[3]
his
is
equi alen
o
equi ing
ha
bo h
A
and
E
be
l.c
.
A-modules
.
5
.
THEOREM
.
The
ollowing
a e
equi alen
dondi ions
on
a
ing
R
(1)
R
is
Vamosian
.
(-2)
R
is
SISI
and
e e y
local
endomo phism ing
is
Vamosian
.
(3)
R
is
SISI
and
e e ylocalinjec i emodule
is
injec i endo
.
PROOF
.

(1)
-
(2)
.
By
Vamos'
heo em,
R
is
SISI,
and
e e ylocal
endomo phism
ing
A
=
End
ER
has
l .c
.
.
injec i e
hull
E
by
(V3)
.
(2)
=
*»(1)
.
Le
E
be
a
local
injec i emodule
o
R,
and
A =
End
RE
.
Since
A
is
Vamosian,
hen
he
injec i e
hull
F
o
i s
uniquesimplemodule
W
is
l
.c
.
o e
A
.
Bu ,
W
C--,
E

and,

in
ac ,
coincides
wi h
he
uniquesimple
R-mod-
11
LEMMA
.
I
P is
a
p ime
ideal
o
R[x],
and
P0
is
he
con ac ed
ideal
in
R,
hen
R[ x]
P
^
Rp
0
[x]Pex
whe e

E
ex

is
he
ex ension
o

P

o

R
P
[x](i
.e
.-Pex=PR
P
[x])
.
O

O
PROOF
.

P
ex

consis s
o
all

g(x)

in

Rp
[x]

wi h
0
coe icien s
in

PR
P
,

and

P
ex

is
p ime
since,

in
gene al,
0
o any
ing
A
and
p ime
ideal
L
o
A,
we ha e
A[x]/L[x]
-
A/L[x]
is
a
domain
.
Le
(x)
=
h(x)/g(x)
deno e
an
elemen
o
he
igh
side,
i
.e
.
Le
h(x),
g(x)
E
Rp
[x],
wi h
g(x)
9P
ex
.
We
can
0
w i e
.h(x)
=h0
(x)/c

and

g(x)
=
g0(x)/d
wi h
c,d
E
R P
0
,
and
g
0 (x),
h
b
(x)
E
R[x]
.
Since
c,d
¢
P0,
hen
-cdg0
q!
P,
hence
h(x)
=h0(x)/cdg
.0(x)
E
R[x]
p
.
The
e e se
inclusion
is

p o ed
simila ly,
i .e
.,
i
h, g
E
R[x],
and
g
¢
P,
hen
we
may
iew
h
and
g
as
elemen s
o
R
p
[x],
0
and
.mo eo e ,
g
¢
P
ex
m
so
h/g
E
R
p
[x]
ex
U
P
12
.
THEOREM
.
.
I
R
is
locally
Noe he ian,
hen
so
is
any
polynomial ingo e
R
in
ini elymany
a iables
x
1
,..
.,x
n
.
In
pa icula ,
hen
R[x1,
. .
.,xn]
is
Vamosian
.

PROOF
.
Since

R

is
locally
Noe he ian,
hen

R
P0
[x]
is
Noe he ian
o any
p imeideal
P
o
R[x],
and
hence,
by
Lemma
13,
so
is
he
local inga
P
.
13
.
COROLLARY
.
I
R
is
on
Neumann egula , hen
R[x1,
. .
.,xn1
is
Vamosian
.
14
.
REMARK
.
R[x]
is
hen
semihe edi a y,
and
con e se-
ly,
i
he
polynomial
ing
R[x]
o e
a
(no
necessa ily
commu-
a i e)
ing
is
semihe edi a y,
hen
R
mus
be
on
Neumann
egula
.
(See [12,
13,
and
14]
.
(Howe e ~gene al,
on
Neumann
egula
ing
R
does
no
imply
R[x]
semihe edi a y)
.
15
PROPOSITION
.
I
R
is
Vamosian
( esp
.
SISI),
hen
so
is
e e y
ac o
ing
.
PROOF
.
I
R
is
SISI,
hen
e e y
ac o
ing
ob iously
is,
so
suppose
ha
R
-is
Vamosian,
and
I
is an
ideal,
and
V
a
simple
R/I
module,
and
E
he
injec i ehullo
V
in
mod-R
.
I
is
easy
o
see
ha
he
annihila o
E
o
I
in
E
is
he
injec i e
hull
o
V
in
mod-R/I
(c
.
he
p oo
o
(2)
=*
(1)
o
Theo em
1)
.
I
ollows
ha
É
is
l
.c
.
o e
R/I,
since
E
is l.c
.
o e
R,
hence
R/I
is
alsoVamosian
.
is
R
.
16
.
COROLLARY
.
I
R[x]
is
Vamosian(SISI),
hen
so
R[x]
is
monic
i
i
con ains
a
monic
polynomial
An
ideal
I
o
R[x]

is
monic
i
I
con ains
a
monicpólynomial,
equi alen ly,
R[xl/I
is
a
ini ely
gene a ed
R-modulé
.
A
ing
R
is
called
a
Mo
nica
ing
i
e e y
co-subdi ec lyi educible
ideal
o
R[x]
is
monic
.
An
ideal
I
o
R
is
colocal
i
R/I
is
a
local
ing
.
Example
.
Any
co-SDI
ideal
I
o
a
SISI ing
is
colocal,
since
R/I
is
hen
indecomposable
injec i e,
hence
has
local
endomo phisin
ing
which
is
isomo phic o
R/I
.
In
his
example
I
is
also
co-PF
in
he
sense
ha
R/I
is PF
.
Thus,
R/I has
a
Mo i a
duali y,
and
hence
R/I
is
Vamos
.
I
P
is
a
subca ego y
o
he
ca ego y
RINGS,
hen
o
any
ideal
H
o
a
ing
A,
we
say
ha
H
is
a
co-P-ideal
i
A/H
E P
.
In
his
pape in e aliawe ha ebeenin e es ed
in
subca ego ies
o RINGSconsis ing
o
:
i educible
(l .e
.
uni o m)
ings,
local
ings,
semilocal
ings,
semipe ec
ings,
sel -
injec i e
ings,
PF- ings,
and
(locally)
Noe he ian
ings
.
17
.
Theo em
.
I
R
is
l
.c
.,
hene e ymonicideal
I
o
R[x]
is
co-semipe ec ,
i
.e
.
o
co-local
ideals
Ii
D
I, i
=

and
>
1
.
Consequen ly,
any
monic
co-i educible
idealo
R[x]
is
co-local
.
R[
x1
/I
=
R[X]/I1x
. . .
XR[
x]
/I
P oo
.
Since
I is
monic,
hen
R[x]/I
is
ini ely
gene a ed
o e
R,
and
hence
by
[1]
o
[3],
is
1 .c
.
as an
R-module
.
By
[161,
any
l
.c
.
ing
is
semipe ec ,
so
R[x]/I
has he
s a ed
decomposi ion
.
18
.
THEOREM
(VAMOS
[2])
.
I
R
is a
Mo i a
ing
(i
.e
.,
has
a
Mo i a
duali y),
hen
so
does
any
algeb a
A
o e
R
ha
is
l.c
.
o e
R,
in
pa icula ,
ha
is a
ini ely
gene a ed
R-module
.
19
.COROLLARY
.
I
R
is
a
Mo i a
ing,
hen R[x]/I
is
a
Mo i a
ing
o
any
monicideal
I,
and
hence R[x]/I
is
sel -injec i e
o
any
monic
co-SDI
ideal
I
.
P oo
.
Ob ious
om
he
aboye
heo em
o Vamos
and
he
p oo
o
Theo em
17
.
20
.
COROLLARY
.
I
R
is
a
MonicaMo i a
ing,
hen
R[x]
is
SISI
.
P oo
.
By
Co olla y
19,
R[x]/I
is
Mo i a,hence
Vamos,
and
he e o e
SISI,
o
e e y
co-SDI
ideal
.
21
.
COROLLARY
.
I
R
is
a l.c
.
VamosianMonica
ing,
hen
R[x]
is
SISI
.
P oo
.
A
ing
R
is
Mo i a
i
R
is l.c
.
and
Vamosian,acco ding
o
Muelle 'sTheo em
s a ed
ea lie
so
R x

is
SISI
by
Co olla y
20
.
22
.
PROPOSITION
.
I
R
is
a
l
.c
.
ing,
hen
he
Muelle -Vamos
conjec u e
implies
ha
R[x]/I
is
Mo i a
o
any
monic
ideal
I
.
P oo
.
Ob ious,
since R[x]/I
is a
ini ely
gene a ed
moduleo e
he
Mo i a
ing
R
.
23
.
COROLLARY
.
I
R
is
a l
.c
.
Monica
ing,
hen
MVC
implies
ha
R[x]
is
SISI
.
Monica?
25
.
REMARK
.
I
R
is
SISI,
i
I
is
a
co-SDI
idealo
R[x]

and
i
I
n
R
is
a
co-SDI
ideal
o
R,
hen
I
can
show ha
I is
monic
i
I
is
co-local
.
This
esul will
appea
elsewhe e
.
P oo
.
Clea om
he
p oo
o
Co ola y
21
.
24
.
QUESTION
.
Is
e e yMo i a,Vámosian,
o
SISI
ing
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Classical
ings,
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34
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114-129
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2
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---------,
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wi h
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3
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Muelle ,
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4
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K
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o
modules
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i s
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o
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83
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I
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21
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C
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Y
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P .J
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.,
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Pillay,
P
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15
.
Vámos,
P
.,
The
decomposi ion
o ini ély
gene a ed
modules
and
ac ionally
sel -injec i e
ings,
J
.
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Soc
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209-220
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16
.
Sandomie ski,
F
.L
.,
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Rebu
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día
12
de
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1986
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The
S a e
Uni e si y,
New
B unswick,
Ni
08903
(USA)
and
Ins i u
d'Es udis
Ca alans
Cen e
de
Rece ca
Ma emá ica
Bella e a,Apa a
50
(SPAIN)