A NNALI DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
A DALBERTO O RSATTI
A class of rings which are the endomorphism rings of some torsion-free abelian groups
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3
esérie, tome 23, n
o1 (1969), p. 143-153
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WHICH ARE THE ENDOMORPHISM RINGS
OF SOME TORSION-FREE ABELIAN GROUPS
ADALBERTO ORSATTI
(*)
Introduction.
A well known result of A. L. S. Corner
([2],
TheoremA)
states that everycountable, reduced,
torsion-freering
A isisomorphic
with the en-domorphism ring E (G)
of somecountable, reduced,
torsion-free group G.(A ring
is called reduced and torsion-free if such is its additivegroup).
In this paper we establish a similar result for a wider class of
rings, precisely
for the class Aconsisting
oflocally countable, reduced,
torsion-free rings.
We say that a torsion-free
ring
A islocally
countable if for everyprime number p
notdividing
A(i.
e.pA ~ A)
thering A/p°°A
iscountable,
where
p°°
A is the intersection of the idealspn
A for every natural numbern.
(Observe
that this definition involvesonly
the additive structure ofA).
The
rings
of class A are characterized as follows(see Proposition 1).
A
ring
Abelongs
to A if andonly
if A isisomorphic
with a puresubring
of a direct
product
IIRp ,
p EP*,
whereP~
is anygiven
set of distinctp
prime
numbersand Rp
acountable, reduced,
torsion-freeZp-algebra.
(Zp
=ring
of rationals whose denominators areprime
top).
The
following generalization
of Theorem A isproved:
THEOREM A~. Let A be a
locally countable, reduced, torsionfree
there exists a
locally countable, reduced,
groupG, of
thesame cardinal as
A,
whoseendoiitorphism ring
E(G)
isisontorphic
with A.Pervenuto alla Redazione il 23 novembre 1968.
(")
Lavoro esegnito nell’ambito delPattiivita dei Gruppi di ricerca del Comitato Na- zionale per la Matematica del C. N. R.The
proof
of this theorem relies on Corner’s methods andideas,
butdoes not make use of Theorem A : in fact it is obtained
by modifying
theproof
of Theorem Aby
means of some localization andglobalization
tech-niques
asexposed
in[5].
It is clear from the above characterization that every
ring
of class L1is of cardinal
2No,
so that onemight
wonder if every suchring belongs
to the class of the
endomorphism rings
ofcountable, reduced,
torsion-free groups. This last class ofrings
has been characterizedby
Corner himself in Theorem 1.1 of[3].
The answer isnegative,
asproved
inProposition 2,
where we construct a
ring
A E A with thefollowing properties : |A | = 2o
andif A = ~
(G),
with G a reduced torsion-free group,then I G ~ > 2No .
It is still an open
question
if therings
of class L1satisfy
thehypotheses
of Theorem 2.2 of
[3],
whichgeneralizes
Theorem 1.1.1.
Preliminaries.
All groups considered in this paper are abelian and
additively
writ-ten ; rings
are associative and with anidentity,
modules areunitary.
IfJ’:
.B -~ A is aring homomorphism, f always
maps theidentity
of B intothe
identity
of A : if B is asubring
ofA, B
contains theidentity
of A.We will
regard
sometimes thering possessing only
the zero element as aring
with anidentity.
Let be an indexed
family
of groups(rings).
We denoteby IIHi
thei
direct
product (=
cartesianproduct)
of theHi and,
for every x EHHi, by
i
xi the
a-component
of x{xi
EEHi
will be thesubgroup (ideal)
ofJ7J?t
i i
consisting
of those elements whosecomponents
are almost all zero.We often attribute to a
ring
someproperties
of its additive group;for istance we say that the
ring
A isreduced, torsion-free, etc... ;
or thata
subring
of A is pure in A.By
asubgroup
of A we mean asubgroup
of the additive group of A.
Let N be the set of
positive integers
and P the set ofprime
numbers(P
cN).
For every group(ring)
~ and for everyp E P, p°°
g will denote the intersection of allsubgroups (ideals) H,
n EEvery
group(ring)
is atopological
group(ring)
in the naturaltopology
obtained
by taking
thesubgroups (ideals) n H, n
EN,
as a basis ofneigh-
bourhoods of 0. This
topology
will be our maintool;
for itsprincipal properties
see[2].
We recall here some of them. Let H be a reduced torsion-free group(ring):
then .g is Hausdorff in the naturaltopology.
LetL be a
subgroup (subring)
of H and endow H with the naturaltopology.
If L is pure in
H,
then the naturaltopology
of L coincides with the re-lative
topology.
L is dense in H if andonly
if the group is divisible.If H is divisible
by
everypositive integer prime
to some fixedp E P,
thenthe natural
topology
of H coincides with thep-adic topology.
The(group
or
ring) homomorphisms
areuniformly
continuousmappings
withrespect
to the natural
topologies
of thecorresponding
structures.Z will denote the
ring
ofintegers, Zp (p
Ep)
thering
of rationals-
whose denominators are
prime
to p,Zp
thering
ofp adic integers. Maps
are written on the left.
The group theoretical
terminology
is that of Fuchs’s book[4].
Let A be a reduced torsion-free
ring.
For everyp E P,
consider thering A~
=(A/p°° A) Q9 Zp (tensor product
ofZ-algebras)
and thering
homo-morphism
A --~A~
resultant of the canonical maps A - A and is torsion-free and without elements(=)=0)
of infi-nite
p-height ;
then the mapA/p°°
A --~~4.~
isinjective
- so that thekernel of g~~ is
p°° A -
andA;
is a reduced torsion-freeZp-algebra.
Define
A *
= IIA: , p E P,
and let A -A*
be the canonical mapgiven by
p
(a)p
= g(a) (a
EA, p E P)
where g
(a)p
is thep-component (a).
Then A~ is a reduced torsion-freering and cp
is aring homomorphism.
In
[5]
we defined for everygroup G
the groupsG~"
=77
and the canonicalhomomorphisms
G -G~,
G -6’*. G~t
p
and
G*
were calledrespectively
the localization and the ralpre-co1npletion
of G. Thisterminology
will be used also for A.From the
embedding
lemma of[5]
we obtain thefollowing
LEMMA 1. Let A be a
torsion-flree Iring.
Then the canonicalhomo1norphism
cp: A --~A~ is injective ;
q~(A)
is a puresubring of A *;
thegroup
(A)
isdivisible,
i.(A) is
dense in A~‘ endowed 1-0ith the na-tural
topology.
Denote
by
Ap theimage ofqJ (A)
under the canonicalprojection 2013;
Ap will be called the
p-proiection
of A.From the definition we
get
AP --- cpp(A).
Since(A)
is a di-visible torsion group with trivial
p-primary component ([5],
pag.5)
andAp
istorsion-free,
we haveLEMMA 2. ..Let A be a reduced
torszon free ring.
Then p-pure the puresubring (subgroup) of Ap
coincideswith dense i7a
A*p,
endouJed 10ith thetopology.
10
The natural
topology
of~
coincides with thep.adic topology.
Let~~ p
~~
A*
be the natural(=p-adic) completion
ofA*;
then~
is a reducedtorsion-free
ring
which containsAp*
as a pure and densesubring ([21,
Lemma1.4). Extending by continuity
theZp-algebra
structure of~
becomes- -
P7
pa
Z.-algebra.
MoreoverA*P
is torsion-free overZp ,
y otherwise the additivep
group of
A*P
would contain somecyclic
p-group.p
A*
is a pure and densesubring
of which iscomplete
inp
the natural
topology,
aseasily
verified.By
means of theinjection
cp, weidentify
A withby
Lemma1,
A becomes a pure and densesubring
of
A*.
Then the naturalcompletion
A of A coincides withA*,
hence with.11
(See [51,
P. 5. and Teorema1).
Now thefollowing
pure and densep
inclusions hold :
n
, ,
Let Z = 1I
P,
be the naturalcompletion
of Z andidentify
Zp
~
, ~
~
with the
(pure
anddense) subring
of Zgenerated by
theidentity
of Z.Extending by continuity
the obviousZ-algebra
structure ofA,
A becomesa
Z-algebra.
Theproduct
of an element yr E Zby
an element a E A isgiven by
thefollowing
relations onp-components :
This is an immediate consequence of the
principle
of the extension of iden-tities, [1],
because(2)
holds for a E Z and a E A.We conclude this section with the
following
remark.LEMMA 3. Let L be a p-pure
subgroup of
the reducedtorsion-free Zp-mo-
dule H. Then the group L
(& Zp
iscanonically isomorphic
with the pure groupof H generated by
L.PROOF. Let
Lp
be the puresubgroup
of Hgenerated by
L.Lp/L
is adivisible torsion group with trivial
p-primary component.
Then the cano-nical
isomorphism
is obtained from the exact sequence 0 -~ .L --~Lp-
2013~
Lp/L
--~ 0(where
the maps are the naturalones) by
tensormultiplica-
tion
by Zp .
2.
The proof of Theorema A*.
Let A
(~ 0)
be aring satisfying
thehypotheses
of TheoremA*.
Since A is reduced andtorsion-free,
the above remarkshold,
inparticular
in-clusions
(1)
are verified.Let P* be the non void
subset
of Pconsisting
of theprimes p
suchthat A. If
p
P* we haveA*
= =0,
hence thep-component
of-
every element of A is zero and we may take A
p E P*.
Notethat,
if p E
P*, A*
is countable. pThe first
part
of theproof
is a localization process : for everypEP.
we construct a countable pure
subgroup Gp
ofAp* 6’p D Ap,
whose endo-morphism ring
isisomorphic
withA*;
in thispart
we will followexactly, except
for smalldetails,
theproof
of Theorem A of[2].
For a
given pEP*,
choose inA*p p
a maximalfamily
of ele-ments of
A*P
plinearly independent
over. ZP
Then for every v EA* p
thereexist a non
negative integer nv
and elements suchthat,
inAt,
where almost all the
n§j
vanish. If we takealways
the smallestpossible nv,
,
then v
uniquely
determinesthe n’ ,
sinceAp
is torsion-free overZp .
LetIIp
be the puresubring
ofZp generated by
thesen) (i
EI, v
E SinceA;
iscountable,
so isIIp .
Moreover we haveLEMMA 4.
If
irtwhere
the Yj
are elementsof Zp linearly independent
overZ7p
andthe Vj
Ethen
the Vj
all vanish.The
proof
of this lemma is the same as the one of Lemma 2.1. of[2].
For every v E
A~ ,
choose two elements ap(v), (v)
EZp
such thatthey
all form a
family
which isalgebraically independent
over This is pos- sible becauseAp
is countable andZp
is of trascendencedegree 2No
overIIp .
Let Ap be thep-projection
ofg : ~.~ C dp .
For every u E Ap definethe element
ep (u)
E1; by putting
, -
where
1
p is theidentity
ofA*p
p I and letGp
p be the puresubgroup
ofA*P
ge-l’, p
nerated
by
AP andby
thesubgroups ofA*p. Gp
is a coun-table reduced torsion-free
Z.-module.
NowGp
containsAt :
in factA~
ispure in
A *
and so,by
Lemma2, A*
is the minimal puresubgroup
ofjlp
containing
AP.Moreover,
for every u EAP, Gp
contains the puresubgroup
H
(u)
ofJL~ generated by ep (u)
AP and it iseasily
verified that H(u)
containsA*.
It is now clear thatGp
coincides with the puresubgroup
ofA*
pgenerated by A*p
p and the It follows thatG. A*
=Gp
, ,
p"’ p
and so every
right multiplication
inA*p by
an element ofA*
induces anendomorphism
onGp.
Thesemultiplications
are distinct because~p
con-tains the
identity
ofA*.
We now prove that everyendomorphism
of6"p
is obtained in this way. Let 6 be - an
arbitrary endomorphism
ofGp.
SinceGp
p is pure and dense in py the natural(= p-adic) completion
ofGp
p coin-cides with the padditive group of
A*p . Consequently 6
extends to aZ,)-endo- morphism 6
ofA* ([2],
Lemma1.4).
Let u be anarbitrary
element of AP and consider 6(e. (u)).
We haveby (3):
Now 6 (ep (u)), 6 (1 v), 6 (u)
are elements ofGp
and so,by
the definition ofGp,
there exist E N such thatwhere
the iii
are distinct elements of andSubstituting
from(5)
in(4)
we obtainAs the ap
(iii)
arealgebraically independent
over77p,
from Lemma 4we
get
while the other
b’s,
c’s and d’s all vanish.By
the last two of(5)
andby (6)
we have :Since
A*
is pure inG,,
it and sinceGp
istorsion-free,
ð
(u)
= 116(lp).
So we see that 6 coincides on A P with theright multiplica-
tion p Now
by
Lemma 2 AP is dense inAp,
andA*p
is dense1*1
p p
in
Gp
forGp
is pure inA*p .
It follows that AP is dense inGp
endowedwith the natural
topology. Then, by
theprinciple
of the extension ofiden_
tities, ð
coincides with theright multiplication
on the whole ofGp.
The first
part
of theproof
is nowcomplete.
The second
part
consists inconstructing
a group G with therequired properties by
means of theGp, p E P*, using
alocal-global argument.
For every a E A consider the elements a
(a), (a)
E Z defined asfollows :
if if
Note that ap,
being the p-component of a, belongs
to Define the elements-
e
(a)
E Aby putting
-
where 1 is the
identity
of A.Every
element of A is determinedby
itsp-components with p E P*; by (2), (3)
and(7)
wehave,
for everypEP.
and a E A :
Let G be the pure
subgroup
. of Agenerated by
A and the e(a) A, a
E A.-
G is reduced torsion-free and of the same cardinal as A. In A we have GA =
G,
so that everyright multiplication by
an element of A induces anendomorphism on G ;
thesemultiplications
are distinct because G contains theidentity
of A. In order tocomplete
theproof
of thetheorem,
it suffices to show that everyendomorphism
of G is obtained in this way and G islocally
countable.For this purpose, let us determine the
Hausdorff p-localizations p E P,
and the natural
pre-completion G *
of G. Since G is pure in A we have for everyp E P
If
p P ,
then hencep°° G =
G andGp
= 0.-
Suppose
then p E P* and let E, be the canonicalprojection
of A ontoBp maps every element of A into its
_p-component. By (9), p°° G
consistsof the elements of G whose
p-component
is zero;consequently G/poo
G is-
canonically isomorphic
withwhich,
aseasily verified,
isp.pnre in A*p.
Hence, by
Lemma3,
weidentify Gp*
with the puresubgroup
of1; generated by
Next we prove thatGp = Gp (p E P*),
from which it follows thatG is
locally countable,
becauseGp
is countable.Gp
containsBp (G) which, by
the definition ofG,
containsAp = Bp (A)
andep (ap) Ap
for every a E A.But,
when a runs overA, ap
exhaustsAP ; hence, by
the definition ofGp, Gp.
On the other hand astraightforward
calculation shows thatthis
implies
and soFrom the above remarks it follows :
Now,
let w be anarbitrary endomorphism
of G.By
P. 2. of[5]
w extendsuniquely
to anendomorphism roll
ofG*. (The proof
of P. 2.suggests
away forconstructing ao).
Observe thatif p and q
are distinctprimes
ofP*,
Hom
(Gp, Gq) =
0 becauseGp
is aZp-module, Gq
is aZq~module
and bothare reduced and torsion-free.
Recalling
how we obtained theendomorphisms
of we see that every
endomorphism
of G* = II EP*,
is inducedby
p
a
right multiplication
in Aby
an element of Then cop
coincides on G with the
right multiplication by w* (1) = co (1) E A* n CT.
Ifwe can prove that co
(1)
EA,
the conclusion is reached. Infact,
we willprove that
A*
n G = A(1).
It is clear thatA*
nG ~ A ; conversely,
if+
n n
gE GnA*,
thenmg = b + ~
e(a;) b~
with mz, n EN, b~
EA,
and Z e ==i=1 i=1
=
mg - b
= c EA* .
As for thejp components
forevery p E P*,
we obtainby (8)
But,
asatp ,
cpbelong
to it follows from Lemma 4 that cp = 0 for every p EP*.
This means c = 0 and so mg = b E A.Hence g E A,
becauseA is pure in G.
3.
The rings of class
ifand
anexample.
Let A be the class of the
rings satisfying
thehypotheses
of Theorem.A~".
A may be characterized as follows.PROPOSITION 1. A
ring
Abelongs
to Aif
andonly if
A isisomorphic
with a pure
subring oj a ring of type
R = IIRp,
p EP*,
whereP*
is a setp
of
distinctprimes and, for
everyp E P*, Rp
is a coutable reducedtorsionjree Zp-algebra.
PROOF. The
necessity
followsimmediately
from Lemma 1. Let A be a puresubring
of R : A is reduced and torsion-free. We haveAlp-
A ==
A/ ( p°° R) n
A for every p E P andA =}= pA
if andonly
For everyp E P*, Alp-
A isisomorphic
with a p-puresubring
ofRp
and is countable..
Finally
we show that the class J is not contained in the class of theendomorphism rings
of countable reduced torsion-free groups ; theserings
are characterized
by
Theorem 1.1. of[3].
PROPOSITION 2. There exists in A a
ring
Aof
cardinal2No
such that every reducedtorsion-free
group, whoseendomorphism ring
isisomorphic
withA,
isof
cardinal 2-
PROOF. For
every p E P,
letRp
be a countable puresubring
ofZp
ofrank >
1.Rp properly
containsZp
as a pure and densesubring.
Define(i)
I am indebted to F. Menegazzo for this suggestion.and consider the
subring A
of R :for almost all
p)
where,
asusual,
is thep-component
of a. We have the proper inclusionsit is
easily
verified that A is pure in R sothat, by Proposition 1,
A Ed ~
A is of cardinal
2No .
Let G be reduced torsion-free group such that E
((T) ==
A. For everyp E P
consider theelement 8,
E A such that thep-component
of 8, is theidentity ip
ofRp
whereas the othercomponents
all vanish. As 8p is anidempotent
element ofA,
Gsplits
into the direct sum of theendomorphic images Ep (G)
and(1 - 8p) (G),
where 1 is theidentity
of A. LetGp
be thesubgroup
of Gconsisting
of those elements which are divisibleby
everyprime
different from p ;Gp
is a reduced torsion-freeZp-module.
Since 8p is divisible in(R
and hencein)
Aby
everyprime
differentfrom p,
while1-Ep
is divisibleby
every powerof p,
we have andOn the other hand
Gp
np°°
G = 0 because G is reduced and torsion-free.Hence
~~ ( G ) = Gp , (1- Ep) ( G ) = p°° G
andWe now show that the
endomorphism ring
ofGp
isisomorphic
withRp . By
the directdecomposition (10),
everyendomorphism fl of Gp
extendsto an
endomorphism
of G such Since s.induces the
identity
onGp,
weConversely,
y every ele-ment of Bp A induces an
endomorphism
onGp
and vanishes onp°°
G. It follows thatE (Gp)
isisomorphic
with thering Ep A,
hence withRp. Every
non trivial
endomorphism
ofGp
isinjective :
infact,
sinceGP
is in a naturalway an
Rr-module, 6’p
is a module overRp
=Zp ;
sinceGP
is a torsion-freeI, 11,
groups
Gp
is torsion-free overZp ;
thenGp
is torsion-free overRp .
It is clear
that,
for everyp E P, G,
coincides with the Hausdorffp-loca-
lization
Gp
of G and sp coincides with the canonicalhomomorphism G -- a;.
.
By
means of the Ep,p E P,
we construct the canonicalhomomorphism E
of G inits natnral
pre-completion
andidentify
G with the pure andp
dense
subgroup e ( G )
of G ~.Since,
if pand q
are distinctprimes,
Hom
(Gp, Gq)
=0,
theendomorphism ring
of IIGp
isII Rp .
As A c IIRp,
p p P
every
endomorphism
of G extends to anendomorphism
of 1IGp. Consequently
p
the effect of a E A on g E G is described
by
thefollowing
formulae on thep-components :
It is
easily
verified that G contains ZGp.
Moreover this inclusion is properp
since the
endomorphism ring
ofZ GV
isII Rp ,
whileE (G ) = A
which isp p
_
not
isomorphic
with Then we can find andelement g E G
and an_
p - _ _
infinite subset P of P such that g,
=j=
0 for overyp E P.
Let A be the idealof A
consisting
of all a E A such that ap = 0 ifp ~ = 2N~
becauseConsider the additive
homomorphism
y : A - Gmapping
p
_ _ _a E A into a
(g)
E G. If a EA,
oc~
0, thereexists p
E P such that0 ;
sinceevery
’non
trivialendomorphism
ofGp
isinjective ar (g~) ~ 0 ; by (11)
thisimplies
a(g) ~ 0,
i. e. y isinjective. Hence 2N, .
REFERENCES
[1]
N. BOURBAKI, Topologiegénérale,
Ch. I, Paris, 1961.[2]
A. L. S. CORNER, Every countable reduced torsion-free ring is an endomorphism ring. Proc.London Math. Soc. (3) 13 (1963) 687-710.
[3]
A. L. S. CORNER, Endomorphism rings of torsion-free abelian groups. Proc. Internat. Conf.Theory of Groups, Austral. Nat. Univ. Canberra, August 1965, pp. 59-69, 1967,
[4]
L. FUCHS, Abelian groups, Budapest 1958.[5]
A. ORSATTI, Un lemma di immersione per i gruppi abeliani senza elementi di altezza infinita.Rend. Sem. Mat. Univ. Padova XXXVIII (1967) 1-13.
Seminario Matematico Università di Padova, Italia