Pub
.
Ma
.
UAB
Vol
.
26
N°
2
Juny
1982
RINGS
WITH
CHAIN
CONDITIONS
ON
PROJECTIVE
IDEALS
1 .
In oduc ion
by
JOHN
KOEHL
Depa men
o
Ma hema ics
Louisiana
S a e
Uni e si y
Rebu
el
2
d'oc ub e
del
1981
A
ing
R is
said
o
sa is y
n igh
a
.c
.c(6
.g
.)
p
i
R
sa i~ ies
he
ascending
chain
condi ion
on
( ini ely
gene a ed)p ojec i e
igh
ideals
;
n igh
d
.c.c
.(J
.g
.)P
is
de ined
simila ily
.
In
sec ion
wo we show ha
i
R
sa is ies
igh
d .c .c
.P
and
i
U is a
wo-sided
idealwhich
is
also
a
mini-
mal
p ojec i e
igh
ideal, hen
ei he
U 2 =
0
o
U
2 =
U(2
.3)
.
I
R is
commu a i e
and
U
is
ini ely
gene a ed,
hen
U
2 =
U
and
is
gene a ed
by an
idempo en elemen
(2
.5)
.
We
alsoshow
ha
i
R
sa is ies
igh
a
.c .c .P
o
igh
d .c
.c
.P
hen
e e y
p ojec i e
igh
ideal
is
coun ably
gene -
a ed

(2
.9)
.
The
polynomial
(also
powe
se ies)
ing
in
in-
ini ely
many
a iables
o e
a
ield
a e
examples
o
non-Noe-
he ian
ings
sa is ying
igh
a
.c .c .P
.
The
symbols
EP
a e
used
o
deno e
enough
pnojec i-
ee,me e y
nonze o
igh
ideal
con ains
a
nonze o
p ojec i e
igh
ideal
;
and MEP
deno es
he
condi ion
ha
e e y
nonze-
o
igh
ideal
con ains
a
nonze b
ini ely
gene a ed
p ojec i-
e
igh
ideal
.
2
.
S uc u e
o
Minimal
P ojec i e
Ideals
I
Ris a
commu a i e
ing
sa is yin
d
.c .c .P
hen
R
has
minimal
p ojec i e
ideals
.
The
nex
heo em
desc ibes
how
hese
ideals
a e
ela ed
o
he
o he
p ojec i e
ideals
.
We
use he
ollowing
known
esul
.
2
.1
Lemma
.
Le-
R
and
S
be n
.íngs,
le
.
P
be
a
p ojec í eR-module
and
le
Q
be
an
(R,S)-
bímodule
ha
íe
p ojec í e
ab
an

S-module
.
Then
P
®p
Q
íz a
p ojec í e
S-module
.
P oo
.
See,
o
example,
Fai h
[3,
11
.15,
p
.4301
.
"
2
.2
Theo em
.
Suppode
Q
íz
a
wo-,síded
.ideal
ín a
n
.íng
R
and
íe
am
.Lnímal
pnojee í e
nígh
ideal
.
Then
gí en
any
pnojee í e
n
.ígh
ideal
P
o6
R,
eí hen
PQ
= 0
o
Q
cP
.
P oo
.
Since
P
and
Q
a e
p ojec i e
igh
ideals
and
Q is
an
(R,R)-bimodule,
P
Q
Q
is
p ojec i e
.
Sin-
ce
P
is
p ojec i e
and
Q
is an
idealo
R,
P
®
Q
=
PQ
so PQ
is
also
p ojec i e
.
Bu
PQ
c
P
and
PQ

Q
.
Since
Q is
a
minimal
p ojec i e
igh
ideal
ei he
PQ

0
o
Q=
PQ
l
P
.
.
2 .3
Co olla y
.
1A P
.í
.6
a
m inímal
p ojec í e
nígh
.ideal'_
wh ich
.(
.6 al'~so
a
- wo-bíded
ídeal,
hen
eí hen
P2
=
P
on
P2
=
0
.
"
2
.3-A
Rema k
.
No e ha bo h o
he
possibili ies
men
ioned
in
2
.3
ac ually
occu
.
Fo
example,
le
R=
¡F
0
lF
)F
be he ing o
2x2
lowe
iangula
ma ices
o e
a
ield
F
.
I
is
well
known
ha
R
is
semihe edi a y
.
The
ideal
P =
(o
0)
sa is ies
P 2 =
0
.
Howe e ,
i
R
is
commu a i e
and
P
is
ini ely
gene a ed,
hen
P 2 =
P
(2
.5)
.
2
.4
Rema k
.
G
.
Michle
[8)
has
shown
ha i
P 2 =
P
in
a
le
pe ec
ing
R,
hen
P
=
ReR
whe e
e
is
an
idempo en
in
R
and
is
cen al
modulo
he
adical
J
.
2
.5
Lemma
.
In
a
commu a í e
níng
R
a
mínimum
pnojec
í e
.ideal'_ P
.C6
ídempoxenx
.
Ib P ís
a16o
bíní e1y
gene&
a ed,
hen
p
i6
gene a ed
by
an
ídempo en
el'emen
.
P oo
.
Since
P is
a
p ojec i e
module,
he
dual
ba-
sis
lemma
gua an ees
ha
he e
exis s
a
se
o
elemen s
{pa},
pa
in P,
and
a
se
o
homomo phisms
{
a
}
wi h
a
in
Hom
R
(P,R)
such ha
a
(P
a
)
=
0
o
almos
al]
a,
and
x =
Ep
a
a
(x)

o
each

x

in
P
.

Since
P
C
R,

a
commu a i-
e
ing,
we
see
ha
x =
E
a
(p
a
x)
o
each
x
in
P
.
Thus
P
2
= 0
implies
P =
0
.

Since
P
0
0, we
see
by
2
.3
ha
P
2
=
P
.
I
P
is
íni el
.y
gene a ed, heo em
76,
page
50
o
Kaplansky
[81
says
ha
P
con ains
an
idempo en
ele-
men
.
Since
P
is
a
minimum
p ojec i e,
he
idempo en
ele
men mus
gene a e
P
.a
2
.6
Theo em
.
14
Rís a
commu a i eeemípn ime
n ing
wí h
EP
ha
sa ís6íe,6
d
.c.c
.P,

hen
R
íe
a
d inee
eum
oj
a
b in i e
numben
oj
b¡eld3
.
P oo
.
Since
R
sa is ies
d
.c
.c
.P,
R
has
minimal
p ojec i e

ideals
.

Say

1
1

is

a
minimal

p ojec i e

ideal
.
Then

11

is

simple
Since

R

has

EP,

and

hence

1
1

is

ge-
ne a ed
by
an
idempo en
e
l
.
By
Jacobson
[6,
P oposi ion
1,

p
.65],

1
1

is
a
ield
.

Le

R
1
=

(1

-
e
l
)R
.

Then

R
1
has
EP
and
sa is ies
d
.c
.c
.P
.
Thus
R
1 =
0
o
con ains
a
minimal

p ojec i e

1
2

whe e

1
2

is

a

ield

gene a ed

by

an
idé iipu en

e
2
.

Then

R
2
=

(1-e1-e2
)R

=

0

o

con ains

a

mi
nimal

p ojec i e

13

which

is

gene a ed

by

an

idempo en
.
Since
R
has
d
.c
.c
.P,
R
con ains
no
in ini e
di ec
sum o
p ojec i e
ideals
so
he
abo e
p ocess
mus
e mina e
a e
a
ini e

numbe

o

s eps
.

Hence

R

=E®I
n
,

a

ini e

sum,

whe-
e
each
I
n
is
a
ield
.a
2
.7
Co olla y
.
16 R
ís
a
commu a i e
Noe he ian
mípn ime
n ing
zha
sa ih6íee
d .c
.c
.P,
hen
R
i6
a
dínee
bum
06
a
6 in i e
numben
ob
bíeId3
.
P oo
.
Since
R
is
Noe he ian,
minimal
p ojec i e
ideals
a e
ini ely
gene a ed
and
hence
gene a ed
by an
idempo
en
.
Thus
minimal
p ojec i es
a e
summands
o he ing
.
The
emainde
o he
p oo
is
he
same
as
he
p oo
o
2
.6
.
"
We
can
also
say
some hing
abou
he
p ojec i e
igh
ideals
in a
ing
sa is ying
igh
a .c
.c.P
.
We
i s s a e
a
heo em
due
o
Kaplansky
[6]
.
2
.8
Theo em
.
E eny
pnojee i e
module
le
a
d inec
eum
o6
coun ablygene a ed
modulee
.
"
2.9
Lemma
.
16
R
ea el ee
n gh
a .c .c .P
o a
n igh
d .c .c
.P
hen
e eny
pnojee i e
n igh
Ideal
e
coun ably
gene a
a ed
.
P oo
.
By
Kaplansky's
heo em,
each
p ojec i e
( igh
ideal)
is a
di ec
sum o
coun ablygene a edp ojec i es
.
I
R
sa is ies
igh
a .c .c .P
o
igh
d
.c
.c
.P
he
numbe
o
independen
summands
o
a
p ojec i e igh
ideal
is
ini e
.
Thus
each
p ojec i e
igh
ideal
is a
sum
o
a
ini e
numbe
o
coun ablygene a ed
modules
a ad
hence
.
i
s
coun ably
gene a-
ed
.
3
.
Inhe i ance
P ope ies
o
Chai a
Condi ions
o a
P ojec i es
3
.1
Theo em
.
Le
R
be
any
n ng
.
11
M R
sa e6íes
he
deeeend ng
cha in
cond i ion
o a
(6 in i ely
gene a ed)
pnojee
i e
6ubmodules
hen
each
homomonph c
mage
o6
M
aleo sa e
6 eb
h e
cond on
.
P oo
.

Le

:

M
-
N

be

a a

R-epimo phism
o

igh
R-modules
a ad
le
P 1
DP
2
DP
3
D
.
.
.
be
a
sequence
o
p ojec i e

14
submodules
o
N
.
Then
-1(P1)--->
P
1
=
0
spli s
so
P
1C
->
-1(P
1
)
CM
.

Thus
P 1
sa is ies
he
descending
chain
condi-
ion
on
( ini ely
gene a ed)
p ojec i e
submodules
.
Hence
he
e
exis s
an
n
such
ha
P
n
= P
n+k o
k =
1,2,3,
.
.
.- I
ollows
ha
N
sa is ies
d .c
.c
.
on
( ini ely
gene a ed)
p ojec i e
submodules
.a
We use he
ollowing
heo ems
o
H
.
Bass [11 and
J .E
.
Bjó k
[21
se e a]
imes
in
he
p oo
o he
ollowing
esul
and
in
he nex
sec ion
.
3 .2
Theo em
(Bass)
.
Le
R
be
a
n,ing,

J
í- z
nad
.i
cal
.
Then
he6ollowíng
ane egu
.i alen
:
1)

R

íz
le6
pen6ec
;
.L
.
e
. ,

e eny
lel
R-module
haz
a
pnojec
.i e
co en
.
2)
.
J
.iz
lel
T-n
.ilpo en
and
R/J
.iz
zemíz
.imple
.
3)

A

dínee
l
.im
.i
o 6

pno
j
ee
.i e
le6
R-modulez

.iz
pno
j
ec
.C e
.
4)
R
za íz6íez
zhe
dezcend
.ing
ehaín
eondí
.ion
on
'

pn
.inc
.ipal
nígh
.idealz
.
5) R
haz
no
.injín
.i e
ze a o6
on hogonal
.idempo en- z,
and
e enynonzeno
níghz
R-module
haz
nonze o
zo-
cle
.a
3
.3
Theo em
(Bjó k)
.
A
níng
R
.iz
le6
pen6ec
.i6
and
only
.i6
R
za íz6íez
he
dezcend
.ing
ehaín
eond
.i
.ion
on
6
.in
.i e
ly
gene a ed
nígh
ídealz
.a
*
P ojec i eco e
is
he dual
o
injec i e
hull
.
3
.4
Theo em
.
16 R
ie
a
nígh penjecz
n
.n
g
and
M R
sazíe6íee
a
.c.c
.(
.g)P
hen
each
homomonph ic
image
o6
M
a eo
ea íelíes
. hís
condí íon
.
P oo
.
Le
:
M
->N
be an
R-epimo phism
and
suppo-
se
ha
P
1
C
P
2
C
P 3
C
.
.
.
is an
ascending
chain
o
p ojec i
e
submodules
o
N
.
Le
P =
UP
i .
Then
P
is
la
.
By
Bass
[11,
P is
p ojec i e
.
Thus
-1
(P)
-
P
-->
0
spli s
and
P
embeds
in
-1
(P)
C
M
.
Hence
P
sa is ies
a
.c.c
.(
.g
.)P
and he
abo e
sequence
o
p ojec i e
modules
mus
e mina e
a e
a
ini e
numbe
o s eps
.
I
ollows
ha
N
sa is ies
a
.c .c
.(
.g
.)P
.
"
4
.
S udy
o
Some
Pa icula
Classes
o
Ringsunde
he
Assump ion
o
Chain
Condi ions
on
P ojec i e
Ideals
.
We can
cha ac e ize
semip ime
igh
semihe edi a y
ings
which
sa is y
.
igh
d
.c
.c
. .g.P
.
Mo e
gene ally,
a
ing
in
which
p incipal
igh
ideals
a e
p ojec i e
is
a
níg
h

P
.P
.

níng
.
4.1
Lemma
.

Le
R
be
a
eem ipn
.íme n .gh P.P
.
níng
.
Then
R
&a íslíe6
nígh
d .c .c
. .g
.P

í6
and
on y
í6 R
íe
e
emíeímp e
.
P oo
.
Since
R
is
igh
P
.P
.,
each
p incipal igh
ideal
is
p ojec i e
.
Thus
igh
d .c .c
. .g
.
P
implies
he des
cending
chain
condi ion
on
p incipal
igh
ideals
.
By
Bass'
heo em,
R/J,
J
=
adical
R,
is
semisimple
and
J,
is
le
T-nilpo en
.
I
J
:
A
0,

he e
exis s
a
minimal
igh
ideal
I
C J
.

Now
ei he
12
= 0
o
he e
is an
e
in
I, e
:
0,
and
e2 = e
.
Since
J
is
le
anishing,
J
con ains
no
nonze o
idempo en s
.
Thus
J
= 0
.
Hence
R
is
semisimple
.
The
con e se
implica ion
is
clea Since
semisimple
ings
a e
A inian
.
"
4
.2
Co olla y
.
16
Ríz
a negulan
n igh d.c .c
. .g
.P
hen
R
i3
aemí4 imple
.
P oo
.
Regula
ings
a e
semip ime
ac ,
semihe edi a y)
and hus
by
4 .1
a e
sa is y
igh
d .c
.c.
.g
.P
.
"
n ing
sa íz6yíng
igh
F
.P
.
(in
semisimple
i
hey
üe ini ion
.
A
ing
R
is
said o
sa is y
(a
.c
.c
.
.)®
i
i
con ains
no
in ini e
se o
independen
igh
ideals
.
The
a
.c
.c
.
on
igh
annule s
o
R
is
abb e ia ed
(a .c
.c)
1
.

I
a
ing
sa is ies
bo h
(a .c .c .) ®

and
(a
.c.c
.)l
,
hen
he
ing
is
said
o be
nígh Goldíe
.
We
say
a
igh
R-module
M
is
uní6onm
i
X,Y
non-
ze o

submodules

o

M

implies

X
n
Y
:
A
0
.

A

ing

is

(nígh )
un ilonm
i
R
R
is
uni o m
.
To
p o e
ou
nex
heo em
we need he
ollowing
heo ems
o
A .W
.
Goldie
[4,51
.
4 .3
Theo em
.
A
n
.íng R
.íe
sem
.ípn.íme
n
.ígh
Gold
.íe
and
on1y
,íb
.í b
quo
.íen
n
.íng
Q(R)
ib
semí
.6ímp
.Le
.a
4
.4
Lemma
.
A un
.íbonm
sem,ípn
.íme
n
.ígh
Gold
.íe
n
.íng
R
.íe
a n
.ígh
Une
doma
.ín
.
"
No e
ha
in
his
case,
as
wi h
al]
domains,
igh
a
.c
.c
. .g
.P
implies
a
.c
.c
.
on
p incipal igh
ideals
.
4 .5
Theo em
.
16
R .íá
a
n
.íng
sa
.íebyíng
n
.ígh
d .c .c
.
.g
.P,
hen
R
.ís
a
n,ígh
Une
n
.íng
and
R
=
Q(R)
.
P oo
.
I
a
is
a
egula
elemen
in
R,
bu no in-
e ible,
hen
a
n
R
p ope ly
con ains
a
n+1
R
o
n =
1,2,
3,
. .
.
.

Bu
a
egula
implies

anR-
R

is
p ojec i e
o
each
n,
con adic ing
d
.c
.c
.
.g .P
.
Thus
each
egula
elemen
in
R
is
igh
in e ible
.
Hence
R =
Q(R)
.@
4
.6
Theo em
.
Le
R
be
a
6em
.ípn
.íme
n
.ígh
.
Go
.Ld .íe
n
.íng
.
Then

R
sa ís6íes
n
.íghx
d
.c
.c
.
.g .P
.íb
and
onky
.íb
R

.í6

s
em
.íb
ímple
.
P o
.
I
R
sa is ies
d .c .c
.
.g .P
hen R
=
Q(R)
by
4
.5
.
Thus
by
4
.3,
R
is
semisimple
.
The
con e se
is
clea
.n
A
domain
sa is ying
he
igh
O e
condi ion
is
a
igh
Goldiedomain
.
The
con e se
o
his
ollows
immedia e-
ly
om
he
ollowing
wo
lemmas
p o en
by
A .W
.
Goldie
[4,51
.
4 .7
Lemma
.

11

R

sa íes
.íes

(a
.
c
.c
.)
9

xhen
e eny
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liza ion
o
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2
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J
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E
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