Publicacions
Ma emá iques,
Vol
36
(1992),
401-406
.
BEHAVIOR
OF
COUNTABLYGENERATED
PURE-PROJECTIVE
MODULES
Abs ac
GORO
AzumAYA
Dedica ed
o
he
memo y
o
P o esso
Pe e
Menal
We
i s
p o e
ha
e e y
coun ably
p esen ed
module
is
a
pu e
epimo phic
image
o
a
coun ably
gene a ed
pu e-p ojec i e
mod-
ule,
and
by
using
his
we show
ha
i
e e y
coun ably
gene -
a ed
pu e-p ojec i e
module
is
pu e-injec i e
hen
e e y
module
is
pu e-injec i e,
while
i
in
any
coun ably
gene a ed
pu e-p ojec i e
module
e e y
coun ably gene a ed
pu e-p ojec i e
pu e
submod-
ule
is
a
di ec
summand
hen
e e y
module
is
pu e-p ojec i e
.
Le
R
be a
ing
.
A
le
R-module
M
is
called
pu e-p ojec i e
i
e e y
pu e
epimo phism
on o
M
spli s
.
As
is
well-known,
M
is
pu e-p ojec i e
i
and
only
i
M
is
a
di ec
summand
o
a
di ec
sum
o
ini ely
p esen ed
le
R-modules
.
On
he
o he
hand,
M
is
called
pu e-injec i e
i
o
any
le
R-module
in
which
M
is
embedded
as
a
pu e
submodule
M
is
always
a
di ec
summand
o
i
.
We
call
R
a
le
pu e-semisimple
ing
o
a
ing
o
le
pu e
global
dimension
ze o
i i
sa is ies
he
ollowing
ob iously
equi alen
condi ions
:
(1)
e e y
le
R-module
is
pu e-p ojec i e,
(2)
e -
e y
le
R-module
is
pu e-injec i e,
(3) o
any
le
R-module
M
e e y
pu e
submodule
o
M
is
a
di ec
summand
o
M
.
I
has
been
known
ha
e e y
le
pu e-semisimple
ing
is
a
le
A inian
ing
and
besides
le
pu e-semisimple
ings
a e cha ac e ized
as
hose
ings
R
o
which
he
ollowing
equi alen
condi ions
hold
:
(4)
e e y
le
R-module
is
a
di ec
sum
o
ini ely
gene a ed
submodules,
(5)
e e y
le
R-module
is
a
di ec
sum
o
indecomposable
submodules,
(6)
e e y
indecomposable
le
R-module
is
ini ely
p esen ed
([2],
[3],
[8],
[10])
.
The
au ho
has
hen
shown
in
[1]
ha
e en
i
we
eplace
he
e ms
e e y
and
any
in (1), (2),
(3)
and
(6)
by
he
e ms
e e y coun ably
gene a ed
and
any
coun ably
gene a ed
espec i ely
we
s ill
ha e
condi ions
each
equi alen
o
he
le
40
2

G
.
AzumAYA
pu e-semisimplici y
o
R
.
Now
in his
pape
we
shall
gi e
u he
cha -
ac e iza ions
o
pu e-semisimple
ings
in
e ms
o
coun ably gene a ed
pu e-p ojec i e
modules
.
In
o de
o
con i m
hese
cha ac e iza ions
we
need
o
combine
Simson's
heo em
in
[6]
ha
i
e e y
coun ably
p e-
sen ed
le
R-module
is
pu e-p ojec i e
hen
R
is le
pu e-semisimple
wi h
he
ollowing
c ucial
p oposi ion,
which
e ines
he
known
Wa ield's
heo em
in
[7]
ha
e e y
module
is
a
pu e
epimo phic
image
o
a
di ec
sum
o
ini ely
p esen ed
modules
and
whe e a
module
is
called
coun -
ably
p esen ed
i i is
isomo phic
o
he
ac o
module
o
a
coun ably
gene a ed
ee
module
modulo
a
coun ably
gene a ed
submodule
:
P oposi ion
1
.
Le
M
be
a
coun ably p esen ed
le
R-module
.
Then
he e
exis s
a
le
R-module
P
which
is
a
coun able
di ec
sum
o
ini ely
p esen ed
le
R-modules
and
has
a
pu e
epimo phism
P
-
M
.
P oo
:
Since
M
is
coun ably
p esen ed,
he e
exis
a
(in ini e-)
coun -
ably
gene a ed
ee
le
R-module
F, a
coun ably
gene a ed
submod-
ule
G
o
F
and
an
epimo phism
W
:
F
~
M
whose
ke nel
is
G
.
Le
ul,
u2, u3,
. . .
be
a
coun able
ee
basis o
F
and
l,
2, 3,
. . .
a
coun -
able
gene a o s
o
G
.
Fo
each
posi i e in ege
n,
we
deno e
by
F
n
he
ini ely
gene a ed
ee
submodule
Rul
G
Ru2
®
.
. .
®
Ru
n o
F
and
by
G
n
he
ini ely
gene a ed
submodule
R l
+
R 2
+
- - -
+
R
n o
G
.
Then
00

Ce
Cea ly
we
ha e
F
=
Y~
F
n
and
G
=
E
G
n
.
Mo eo e ,
o
each
n,
he e
n=1 n=1
is
an
m
such
ha
G
n
C
F
n
, .
Le
l(n)
be
he
leas
one
among
such
m's,
and
le
m(n)
=
max(l(n),
n)
.
Then,
i is
ob ious
ha
G
n
C
F,,(
n
)
o
e e y
n
and
F=

Fnn(n)
.
n=1
00
Conside
now
coun able
(ou e )
di ec
sums
S =
®
F,
(n
)
and
T
=
n=1
®
G
n
.

Then
T
is
a
submodule
o
S
.

Fo
each
n,
we
deno e
by qn
n=1
he
n- h
canonical
embedding
F,,( n
)
,
S,
Le
.,
qn(x),
o
x
E
F,,( n
),
is
he
elemen
o
S
whose
n- h
en y
is
x and
o he
en ies a e
all
0
.
Clea ly
he
es ic ion
o
q
n
o
he
submodule
G
n
o
,,,(n)
is
he
n- h
canonical
embedding
G
n
~
T
.
Now
he
ac o
module
F,)/G,
is
ini ely
p esen ed
o
each
n
.
Le
P=
®(F,

,(n)/Gn)
be
he
coun able
n=1
di ec
sum
.
Then
he
na u al
epimo phisms
F

,
(n
)
-
Fm(n)/Gn
oge he
de ine
an epimo phism 0
:
5
-->
P
whose
ke nel
is
T
.
I
we
nex
associa e
COUNTABLY
GENERATED
PURE-PROJECTIVE
MODULES

403
each
s
E
S
wi h
he
sum
n-1
o
s,
hen
we
ha e
an epimo phism
:
S

F
.
The
es ic ion o
o
G
gi es
clea ly
an
epimo phism
g
:
T
->
G,
and
so
we
ha e
ha
cp(
(T))
=
W(g(T))
=
cp(G)
=
0
.
Since
T
is
he ke nel
o
7P,
his
implies
ha
an
epimo phism
h
:
P

M
is
well-de ined
so
ha
h
o
0
=W
o
.
We
now
show
ha
h
is
a
pu e
epimo phism
.
Le
E
be a
ini ely
p esen ed
le
R-module
.
Thus
he e
exis
a
ini ely
gene a ed
ee
le
R-module
L
and
an epimo phism
7
:
L
-->
E
whose
ke nel
K
is
ini ely
gene a ed
.
Le
ca
:
E
->
M
be a
homomo phism
.
Conside
he
p oduc
a
o
7
:
L
---~
M
.
Since
L
is
p ojec i e,
he e
mus
exis a
homomo phism
,l
:
L
-->
F
such
ha
cW
o
~3
=
ca
o
7
.
F om
his
ollows
ha
W(O(K))
=
a(7 (K))
=
0
and
so
O(K)
C
G,
he
ke nel
o
cp
.
Since
bo h
L
and
K
a e
ini ely
gene a ed,
hei
homomo phic
images
~3(L)
and
O(K)
a e
ini ely
gene a ed
submodules
o
F
and
G
espec i ely
.
The e o e
we
ha e
(L)
C
F,(n)
and
/3(K)
C
G
n
o
su icien ly
la ge
n
.
We
ix
such
an
n,
and
de ine
he
homomo phism
-y
:
L
-
S
by
y=
q
n
o
/3
.
Then
we
ha e
o
y
=
o
qn
o
~3,
bu
Since
o
qn
is
clea ly
he
inclusion
map
F,(n)
---+
F
i
ollows
ha
o
y
=
Q
.
On
he
o he
hand,
ha
/3(K)
C
G
n
implies
ha
y(K)
=
g
n
(O(K))
C
g
n
(G
n
)
C
T
and
he e o e
0(y(K))
C
O(T)
=
0,
Le
.,
K
is
con ained
in
he ke nel
o
0
o
y
:
L
-~
P
.
This
shows
ha
a
homomo phism
5
:
E
-->
P
is
well-de ined
so
ha
6o7
=
Ooy
.
We
ha e hen
hobo7
=
ho0oy
=
cpo
oy
=
cpoo
=
ao7 ,
bu
Since
n
:
L
->
E
is
an
epimo phism
we know
ha
h
o
6
=
ca
.
This
is
ue
o
e e y
ini ely
p esen ed
module
E
and
o
e e y
homomo phism
a
:
E
-
M,
and
hus
i is
p o ed
ha
h
is
a
pu e
epimo phism
.
s
nin
F, whe e
s
n
(6F,,(
n
»
is
he
n- h
en y
Theo em
2
.
R
is
le
pu e-semisimple
i
and
only
i
e e y coun ably
gene a ed
pu e-p ojec i e
le
R-module
is
pu e-injec i e
.
P oo
..
Clea ly
we
need
only
p o e
he
i
pa
.
Le
M
be any
coun -
ably
p esen ed
le
R-module
.
Then
by
he
p eceding
p oposi ion
he e
exis
a
coun able
di ec
sum
P
o
ini ely
p esen ed
le
R-modules and
an epimo phism
h
:
P
--->
M
whose
ke nel
Q
is
a
pu e
submodule
o
P
.
Now
he
di ec
sum
P(`
),
N
being
he
se
o
all
na u al
numbe s,
o
he
coun able
numbe
o
copies
o
P
is
also a
coun able
di ec
sum
o
ini ely
p esen ed
le
R-modules
and
hence
is
coun ably gene a ed
and
pu e-p ojec i e
.
Thus
by
ou
assump ion
p(N)
is
pu e-injec i e,
o
equi alen ly,
P
is
E-pu e-injec i e
.
The e o e
he
pu e
submodule
Q
o
P
is
a
di ec
summand
o
P
([9,
p
.
1100],
[1,
P op
.
3
.5]),
which
means
ha
M
can
be
embedded
in o
P
as
a
di ec
summand
and
he e o e
M
is
pu e-p ojec i e
oo
.
Thus
i
u ns
ou
ha
e e y
coun ably
p esen ed
40
4

G
.
AzumAYA
le
R-module
is
pu e-p ojec i e
.
I
ollows
om Simson
[6,
Th
.
6
.3]
ha
R
is
le
pu e-semisimple
.
As
is
easily
seen,
e e y
coun able
di ec
sum
o
coun ably
p esen ed
modules
is
coun ably
p esen ed
oo
and
so
in
pa icula
e e y
coun able
di ec
sum
o
ini ely
p esen ed
modules
is
coun ably
p esen ed
.
The e-
o e,
om
he
p eceding
heo em
we
can
de i e
he
ollowing,
which
is
howe e
ega ded
as
a
dual o
he
aboye
e e ed
Simson's
heo em
:
Co olla y
3
.
R
is
le
pu e-semisimple
i
and
only
i
e e y coun ably
p esen ed
le
R-module
is
pu e-injec á e
.
Now
he
ollowing
is
o
e ine
he
equi alen e
o
he
condi ions
(2)
and
(6)
in
[1,
Th
.
4
.5]
:
Theo em
3
.
R
is
le
pu e-semisimple
i
aud
only
i
o
any
coun -
ably
gene a ed
pu e-p ojec i e
le
R-module
P
e e y
coun ably
gene a ed
pu e-p ojec i e
pu e
submodule
o
P
is
a
di ec
summand
o
P
.
P oo
.
:
We
need
only
p o e
he
i
pa
oo
.

Le
M
be a
coun ably
p esen ed
le
R-module
.
This
means
ha
he e
exis
a
coun ably
gene -
a ed
ee
le
R-module
F
and an
epimo phism
cp
:
F
-->
M
whose
ke nel
G
is
coun ably
gene a ed
.
On
he
o he
hand,
by
P oposi ion
1,
he e
exis
a
coun able
di ec
sum
P
o
ini ely
p esen ed
le
R-modules
and
an
epimo phism
h
:
P
-->
M
whose
ke nel
Q
is
pu e
in
P
.
We
shall
show
ha
Q
is
coun ably
gene a ed
oo
.
Fo ,
since
F
is
p ojec i e
he e
is
a
homomo phism
:
F
-->
P
such
ha
h
o
=
(p
.
Le
p
be
an
elemen
o
P
.
Then
h(p)
is
in
M
and
so
we
can
ind
an x E
F
such
ha
W(x)
=
h(p)
whence
h(
(x))
=
h(p),
Le
.,
h(p-
(x))
=
0
.
Thus
weknow
ha
p-
(x)
is
in
Q
and
he e o e
ha
P
=
(F)
+Q
.
On
he
o he
hand,
i
x
is
in
F
hen
he
equali y
h(
(x))
=
cp(x)
implies
ha
(x)
is
in
Q
i
and
only
i
x
is
in
G,
and
he e o e
we
ha e
(F)
1
Q=
(G)
.
Thus
we know
ha
Q/
(G)
--
P/
(F)
.
Bu
since
P
is
coun ably
gene a ed
i s
homomo phic
image
P/
(F)
whence
Q/
(G)
is
coun ably
gene a ed
oo,
while
since
G
is
coun ably gene a ed
i s
homomo phic
image
(G)
is
also
coun ably
gene a ed
.
F om
his
we
can
conclude
ha
Q
is
coun ably gene a ed
.
Now,
since
P
is
pu e-p ojec i e,
he
coun ably
gene a ed
pu e
submod-
ule
Q
o
P
is
pu e-p ojec i e
acco ding
o
Kielpi íski-Simson
[4,
Co
.
1
.5]
.
(This
can
also
be
p o ed
by
using
he
no ion o
Mi ag-Le e
modules
as
ollows
:
As
is
well-known,
e e y
pu e
submodule
o
a
pu e-p ojec i e
module
is
a
Mi ag-Le ie
module,
and
so
Q
is
a
Mi ag-Le ie
mod-
ule,
Le
.,
he canonical
homomo phism
( l
A
Z )
~
Q
,
11
(A
¡
®
Q)
is
R

R
COUNTABLY
GENERATED
PURE-PROJECTIVE
MODULES

405
a
monomo phism
o
e e y
amily
{Ai}
o
igh
R-modules
.
Since
Q
is
coun ably
gene a ed,
Q
mus
be
pu e-p ojec i e
by
Raynaud-G uson
[5,
Pa
II,
Co
.
2
.2 .2]
.)
By
he
assump ion
o
ou
heo em,
we know
ha
Q
is
a di ec
summand
o
P
and
so
M
can
be
embedded
in o
P
as
a
di ec
summand,
which
implies ha
M
is
pu e-p ojec i e
.
Thus
we
ha e
shown
ha
e e y
coun ably
p esen ed
le
R-module
is
pu e-
p ojec i e,
and
he e o e
again
by
Simson's
heo em
([6,
Th
.
6
.3])
R
is
le
pu e-semisimple
.
In his
connec ion,
i is
o be
poin ed
ou
ha
Theo em
2
is
a
di-
ec
consequence
o
Theo em
3
.
Fo ,
le
P
be a
le
R-module
and
Q
a
coun ably
gene a ed
pu e-p ojec i e
pu e
submodule
o
P
.
Suppose
ha
e e y
coun ably
gene a ed
pu e-p ojec i e
le
R-module
is
pu e-
injec i e
.
Then
Q
is
a di ec
summand
o
P
.
Thus,
by
Theo em
3,
R
is
le
pu e-semisimple
.
Rema k
.
A
.
Abe
has
independen ly ob ained
P oposi ion
1
and
The-
o em
2
oo
.
Indeed,
he
p o es P oposi ion
1
by
using
[6,
Co
.
2
.5]
ha
i
a
module
M
is
a
di ec
limi
o
modules
M
i
's
hen
he
canonical
epimo phism
®
M
i -->
M
is
pu e
.
i
Re e ences
1
.

G
.
AzumAYA,
"Coun able
gene a edness
e sion
o
ings
o
pu e
global
dimension
ze o,"
Rep esen a ions
o
algeb a,
Camb idge
Uni-
e si y
P ess,
1992
.
2
.

S
.
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