Publ. Ma . 47 (2003), 359–372
CHARACTERIZATION OF THE INESSENTIAL
ENDOMORPHISMS IN THE CATEGORY OF ABELIAN
GROUPS
S. Abdelalim and H. Essannouni∗
Abs ac
An endomo phism o an Abelian g oup Ais said o be inessen-
ial (in he ca ego y o Abelian g oups) i i can be ex ended o
an endomo phism o any Abelian g oup which con ains Aas a
subg oup. In his pape we show ha is as abo e i and only
i ( − idA)(A)iscon ained in he maximal di isible subg oup
o A o some ∈Z.
1. In oduc ion
Th oughou his pape , we will ollow he e minology o [2]. Le M
be an objec o a ca ego y Cand ∈End(M), is called inessen ial
(in C)i o any monomo phism σ:M→N he e exis s ˜
∈End(N)
such ha ˜
σ =σ ,ino he wo ds he ollowing diag am
Mσ
−−−−→N





˜
Mσ
−−−−→N
commu es.
Ines(M) deno es all he inessen ial endomo phisms o M.Mis called
igid i End(M)=Ines(M). Fo a conc e e ca ego y C, he cha ac e -
iza ion o he inessen ial endomo phisms is one o he p oblems aised
in [2]. In his pape , we ake C=Ab he ca ego y o he Abelian g oups
and we show o an Abelian g oup A, and an endomo phism o A,
ha is inessen ial (in Ab)i and only i he e exis s ∈Zsuch ha
2000 Ma hema ics Subjec Classifica ion. 20K30.
Key wo ds. Ca ego y, inessen ial, monomo phism, igid, di isible, ex ension.
∗The second au ho is pa ially suppo ed by he Minis e ´ıo de Cienc´ıa y Tecnolog´ıa.
P oyec o BEM 2001–2335, and he hanks P o esso A. Kaidi o he help ul con e -
sa ions du ing his isi o he Uni e si y o Alme ´ıa.
360 S. Abdelalim, H. Essannouni
( − idA)(A)⊆D, whe e Dis he maximal di isible subg oup o A.In
pa icula i Ais educed hen Ines(A)=ZidA. The p oo o his esul
uses he p ope ies o he endomo phisms o some ex ensions o ce ain
di ec sums o o sion cyclic g oups.
F om now on, he wo d g oup means Abelian g oup and we adop he
no a ions o [3].
2. Some cons uc ions
Cons uc ion 1. Le (αn)n≥0beasequence o na u al numbe s such
ha αn<α
n+1 and 2αn+1 −αn+n+3≤αn+2,∀n∈N.I wepu
θn=αn−αn−1−n o n≥1 hen we ha e θn−θn−1≥n,n≥2. Le
p∈N∗and ( n,m)n≥mbease o nonze o na u al numbe s, ela i ely
p ime wi h psuch ha i,j j,k = i,k i i≥j≥k.
We conside he di ec p oduc 
n≥1
xnwi h o(xn)=pαnand deno e
by ϕk:
n≥1
xn→xk he canonical p ojec ion. Fo m≥1, we define
he elemen gmo 
n≥1
xnby
ϕn(gm)=0i n<m
pαn−αmxni n≥m.
We di ec ly check ha o(gm)=pαm,xm=gm−pαm+1−αmgm+1 and
{gm/m ≥1} =
m≥1
gm.
Le m∈N∗and ξa unc ion om Nin o {0,1},wedefine he ele-
men S(m, ξ)o 
n≥1
xnby
ϕn(S(m, ξ)) = 0i n<m
ξ(n) n,mpn−m+αn−1xni n≥m.
We ha e
S(m, ξ)=

n=m
ξ(n) n,mpn−m+αn−1xn+ +1,mp +1−mS( +1,ξ)
i ≥m.
Le K1be he subg oup o 
n≥1
xngene a ed by
{gm/m ≥1}∪{S(m, ξ)/m ≥1,ξ∈{0,1}N}.
Inessen ial Endomo phisms in he Abelian G oups 361
Lemma 2.1. The di ec sum 
n≥1
xnis a subg oup o K1and o all
λ∈End(K1) he e exis s, N ∈Nand ∈Zsuch ha s,1pαn−nλ(xn)=
pαn−n xn,∀n≥N.
P oo : Le λ∈End(K1). Le us show a fi s ha he e exis s N0≥1
such ha i n>m≥N0 hen ϕn(pαm−mλ(xm)) = 0.
I no , we can find a sequence (mk)k≥1such ha o all k≥1, he e
exis s nk>m
kwi h ϕnk(pαmk−mkλ(xmk)) =0and αnk≤mk+1. Le
ζ:N→{0,1}be he unc ion defined by ζ(n)=1i n∈{mk/k ≥1}
and ζ(n)=0o he wise. We can w i e:
λ(S(1,ζ)) =
a

i=1
cigi+
b

j=1
djS(m, ξj).
I we pu =αa, hen p λ(S(1,ζ)) = p b

j=1
djS(m, ξj). Fo any k,we
ha e
pθmk+1S(1,ζ)=pθmk+1
×mk+1−1

n=1
ζ(n) n,1pn−1+αn−1xn+ mk+1,1pmk+1 S(mk+1,ζ)∈pαnkK1
because θmk+1+n−1+αn−1≥αni mk≥n≥1, ζ(n)=0i
mk+1 >n>m
kand θmk+1+mk+1 ≥αnk.I kis la ge enough, hen
ϕnk(pθmk+1λ(S(1,ζ))) = ϕnk
pθmk+1
b

j=1
djS(m, ξj)
=0
he e o e pθnk−θmkdi ides (nk), whe e (n)=
b

j=1
djξj(n). Since he
se { (n)/n ∈N}is fini e and θnk−θmk≥nk, hen he e exis s k1≥1
such ha (nk)=0,∀k≥k1.On he o he hand
pθmk−mk+1S(1,ζ)− mk,1pαmk−mkxmk∈pαnkK1,
he e o e
ϕnk
pθmk−mk+1
b

j=1
djS(m, ξj)
=0
o kla ge enough. The e o e i exis s k2≥1 such ha (nk)=0, ∀k≥k2,
which is absu d. Thus he e exis s N0∈Nsuch ha : pαn−nλ(xn)=
362 S. Abdelalim, H. Essannouni
pαn−n nxn,∀n≥N0, whe e n∈Z. Since T(K1)= 
m≥1
gmand
αk≤αn−n o k<n, he e o e pαn−nλ(gn)∈pαn−n(
k≥n
 gk). Le
m≥N0and pu o n≥m,un=li pαn−mλ(gn)=pαn−ml

k=n
kgk
wi h (pαn−m lgl=0and l>n) and un=0i pαn−mλ(gn)∈pαn−mgn.
Since xn=gn−pαn+1−αngn+1,i iseasy o see ha he sequence (un)n≥m
is dec easing. Since o un=0weha e un>n, hen he e exis s
Mm≥msuch ha un=0,∀n≥Mm. The e o e pαn−mλ(gn)∈
pαn−mgn,∀n≥Mm. Le ξ0(n)=1,∀n∈N.Wecan w i e:
pk

λ(S(1,ξ
0)) = pk
k

j=1
mjS(s, ξj), whe e k,k,s ∈N,m1,...,m
k∈Z
and ξ1,...,ξ
k∈{0,1}N.
We ha e pθn−n+1S(1,ξ
0)− n,1pαn−nxn∈pαnK1 hus o nla ge
enough n,1pαn−nϕn(λ(xn)) = pθn−n+1ϕn(
k

j=1
mjS(s, ξj)) =⇒pn+s−1
di ides s,1ps−1 n−w(n) whe e w(n)=
k

j=1
mjξj(n). Acco dingly, i
d∈Zsuch ha he se {n∈N/w(n+1)−w(n)=d}is infini e, hen pm
di ides d,∀m≥N0, he e o e d=0. Since he se {w(n+1)−w(n)/n ∈
N}is fini e, hen he e exis 0∈Zand N1∈Nsuch ha w(n)= 0,
∀n≥N1.I isclea ha ps−1di ides 0. Finally i we pu 0=ps−1 ,
we can find N∈Nsuch ha s,1pαn−nλ(xn)=pαn−n xn,∀n≥N.
Cons uc ion 2. Le (αn)n≥0be as in Cons uc ion 1, and le pand
qbe wo na u al numbe s diffe en om ze o and ela i ely p ime, we
conside he wo di ec p oduc s

n≥1
xnand 
n≥1
ynwi h o(xn)=pαnand o(yn)=qαn,∀n≥1.
The elemen s hmo 
n≥1
yna e defined in he same way as he gm
o 
n≥1
xn(see Cons uc ion 1). The elemen s S1(m, ξ) ( espec i ely
S2(m, ξ)) o 
n≥1
xn( espec i ely 
n≥1
yn) a e defined like S(m, ξ)o
Cons uc ion 1 wi h n,m =qn−m( espec i ely n,m =pn−m).
Inessen ial Endomo phisms in he Abelian G oups 363
We pu R(m, ξ)=S1(m, ξ)+S2(m, ξ)∈(
n≥1
xn)⊕(
n≥1
yn), hen
we ha e,
R(m, ξ)=

n=m
ξ(n)(pq)n−m(pαn−1xn+qαn−1yn)+(pq) +1−mR( +1,ξ)
i ≥m.
Le K2be he subg oup o ( 
n≥1
xn)⊕(
n≥1
yn) gene a ed by
{gm/m ≥1}∪{hm/m ≥1}∪{R(m, ξ)/m ≥1,ξ∈{0,1}N}.
Lemma 2.2. The di ec sum (
n≥1
xn)⊕(
n≥1
yn)is a subg oup o
K2and o all λ∈End(K2), he e exis ∈Z,N∈Nsuch ha
pαn−nλ(xn)=pαn−n xnand qαn−nλ(yn)=qαn−n yn,∀n≥N.
P oo : Le µ:(
n≥1
xn)⊕(
n≥1
yn)→
n≥1
xnbe he canonical p o-
jec ion. Then µ(K2)is he g oup K1o Cons uc ion 1 (wi h n,m =
qn−m). Le λ∈End(K2). The e exis s λ1∈End(µ(K2)) such ha
λ1(µ(X)) = µ(λ(X)), ∀X∈K2.ByLemma 2.1 he e exis s1,N
1∈N
and 1∈Zsuch ha qs1pαn−nλ1(xn)=pαn−n 1xn,∀n≥N1, he e-
o e qs1pαn−nλ(xn)=pαn−n 1xn,∀n≥N1.In he same way he e a e
s2,N
2∈Nand 2∈Zsuch ha ps2qαn−nλ(yn)=qαn−n 2yn,∀n≥N2.
We can ake s1=s2=sand N1=N2=N. Le ξ0(n)=1,∀n∈N,we
can w i e: (pq)lλ(R(1,ξ
0)) = (pq)lk

j=1
mjR(m, ξj) whe e l,k,m ∈N∗,
m1,...,m
k∈Zand ξ1,...,ξ
k∈{0,1}N.Wecan ake m≥1+s.By
applying µ o his equali y, we ob ain:
plλ(S(1,ξ
0)) = pl
k

j=1
mjS(m, ξj).
Then o nla ge enough pn+m−1di ides qm−1−spm−1 1− (n) whe e
(n)=
k

j=1
mjξj(n) (see he p oo o Lemma 2.1). Le d∈Zsuch ha
he se {n∈N/ (n)=d}is infini e, hen d=qm−1−spm−1 1in he
same way d=pm−1−sqm−1 2.I wepu 1=qs and 2=ps , hen we
can find N∈Nsuch ha
pαn−nλ(xn)=pαn−n xnand qαn−nλ(yn)=qαn−n yn,∀n≥N.

364 S. Abdelalim, H. Essannouni
Cons uc ion 3. Le (αn)n≥0be as in Cons uc ion 1 and (βn)n≥1be
a sequence o nonze o na u al numbe s. Le p, q1,...,q
n,... be nonze o
ela i ely p ime na u al numbe s. Le us conside he g oup ( 
n≥1
xn)⊕
(
n≥1
zn) wi h o(xn)=pαnand o(zn)=qβn
n,∀n≥1, he elemen s gm
and S(m, ξ)o 
n≥1
xna e defined as in Cons uc ion 1 wi h
n,m =












1i n=m
q1···qmi n=m+1
(q1···qm)n−mn−m−1

j=1
qn−m−j
m+ji n≥m+2,
he elemen R(m, ξ)o 
n≥1
znis defined as ollows
ϕn(R(m, ξ)) = 


0i n<m
ξ(n)pn−m n,mzni n≥m
whe e ϕk:
n≥1
zn→zkis he canonical p ojec ion. I we pu
T(m, ξ)=S(m, ξ)+R(m, ξ)∈

n≥1
xn)⊕(
n≥1
zn
,
we ha e
T(m, ξ)=

n=m
ξ(n) n,mpn−m(pαn−1xn+zn)+ +1,mp +1−mT( +1,ξ),
i ≥m.
Le K3be he subg oup o ( 
n≥1
xn)⊕(
n≥1
zn) gene a ed by {gn/n ≥
1}∪{zn/n ≥1}∪{T(m, , ξ)/m ≥1,ξ∈{0,1}N}.
Lemma 2.3. The di ec sum (
n≥1
xn)⊕(
n≥1
zn)is a subg oup o K3
and o all λ∈End(K3), he e exis ∈Zand N,s ∈Nsuch ha
s,1pαn−nλ(xn)=pαn−n xnand s,1λ(zn)= zn,∀n≥N.
P oo : Le µ:(
n≥1
xn)⊕(
n≥1
zn)→
n≥1
xnbe he canonical p o-
jec ion. Then µ(K3)=K1is he g oup o Cons uc ion 1. Le λ∈
End(K3), he endomo phism λ1o K1defined by λ1(µ(X)) = µ(λ(X)),
Inessen ial Endomo phisms in he Abelian G oups 365
∀X∈K3,iswell defined. Acco ding o Lemma 2.1 he e exis s, N0∈N
and ∈Zsuch ha s,1pαn−nλ1(xn)=pαn−n xn,∀n≥N0.I is
clea ha λ(zn)∈zn,∀n≥1. Pu ing λ(zn)=knzn,∀n≥1,
we conside ξ0:N→{0,1}wi h ξ0(n)=1,∀n∈Nwe can w i e:
pl λ(T(1,ξ
0)) = pl
k

j=1
djT(m, ξj) whe e and pa e ela i ely p ime
and m≥s.Byapplying µ o his equali y, we ob ain:
plλ1(S(1,ξ
0)) = pl
k

j=1
djS(m, ξj).
Following he same s eps as in Lemmas 2.1 and 2.2, we can find N1∈N
such ha pn+m−1di ides m,spm−1 − (n), ∀n≥N1,wi h (n)=
k

j=1
djξj(n). Then he e exis s N2such ha (n)= m,spm−1 ,∀n≥N2.
I n≥m, hen qβn
ndi ides m,1pm−1kn− (n). Finally he e exis s N∈N
such ha s,1pαn−nλ(xn)= pαn−nxnand s,1λ(zn)= zn,∀n≥N.
3. Cha ac e iza ion o he inessen ial endomo phisms in
he ca ego y o he Abelian g oups
In he ollowing, we suppose ha Ais a g oup, and an endomo -
phism o Asa is ying he ollowing p ope y.
(E): Fo any exac sequence 0 →Aσ
→B he e exis s ˜
∈End(B) such
ha he ollowing diag am
0−−−−→Aσ
−−−−→B





˜
0−−−−→Aσ
−−−−→B
is commu a i e.
Le (αn)n≥0be a sequence as in Cons uc ion 1.
Lemma 3.1. Fo all a∈Aand any q∈N∗, he e exis s ∈Zsuch
ha ( (a)− a)∈
n≥0
qnA.
P oo : Le us conside he ee g oup L=
n≥1
en.Wepu G=A⊕L,
G0={a−qαnen/n ≥1} and G=G/G0. The homomo phism σ:A→
Gdefined by σ(b)=b+G0is a monomo phism, and i xn=en+σ(A)
(en=en+G0) hen G/σ(A)= 
n≥1
xnand o(xn)=qαn,∀n≥1. Le
366 S. Abdelalim, H. Essannouni
K1beasubg oup o 
n≥1
xndefined in Cons uc ion 1 (wi h n,m =1,
∀n≥m). The e exis s a commu a i e diag am, whose ows a e exac ,
and which has he ollowing o m:
0−−−−→A−−−−→Gπ
−−−−→G/σ(A)−−−−→0







0−−−−→Aσ
−−−−→Bµ
−−−−→K1−−−−→0
(see [3, 24.6]). We can find ˜
∈End(B) and λ∈End(K1) such ha
˜
σ =σ and λµ =µ˜
.ByLemma 2.1, he e a e ∈Zand N∈Nsuch
ha qαn−nλ(xn)= qαn−nxn,∀n≥N.Fo n≥N,µ[qαn−n(˜
(en)−
en)] = 0, he e o e ( (a)− a)∈qnA,so( (a)− a)∈
n≥0
qnA.
Co olla y 3.2. I A1=0, hen o all a∈T(A) he e exis s a∈Z
such ha (a)= aawhe e T(A)is he o sion pa o A.
P oo : Le us pu q=o(a) and le ∈Zsuch ha ( (a)− a)∈
n≥0
qnA.
Le pbeap ime numbe , i pdi ides q hen ( (a)− a)∈
n≥0
pnAand
i pand qa e ela i ely p ime, we also ha e ( (a)− a)∈
n≥0
pnA, hus
(a)= a.
Lemma 3.3. I A1=0, hen he e exis s ∈Zsuch ha (a)= a,
∀a∈T(A).
P oo : We suppose ha T(A)isbounded, hen he e exis s x0∈T(A)
such ha x0is a di ec summand o T(A) and o(x0).T(A)=0. I
(x0)= x0, hen ∀a∈T(A), (a)= a.Wenow suppose ha T(A)is
no bounded. I pis p ime numbe , we deno e by Tp he p-componen
o T(A).
1s case: The e exis s a p ime numbe psuch ha Tpis no bounded.
Le Sbe a basic subg oup o Tp,wecan w i e
S=

n≥1
an
⊕S0wi h o(an)=p nand 1 ≤ n<
n+1,∀n≥1.
Inessen ial Endomo phisms in he Abelian G oups 367
Fo each n≥1, we conside anas an elemen o he g oup Xnwi h
pαnXn=an. The e exis s a g oup Gsuch ha :
A≤G,


n≥1
Xn
≤G,
A+

n≥1
Xn
=G
and
A∩

n≥1
Xn
=
n≥1
an.
We pu xn=Xn+A, hen G/A =
n≥1
xnand o(xn)=pαn,∀n≥1.
By [3, P oposi ion 24.6], he e exis s a commu a i e diag am, whose
ows a e exac , and has he ollowing o m:
0−−−−→A−−−−→Gπ
−−−−→G/A −−−−→0







0−−−−→Aσ
−−−−→Bµ
−−−−→K1−−−−→0
.
K1is he g oup o Cons uc ion 1 (wi h n,m =1,∀n≥m). The e a e
˜
∈End(B) and λ∈End(K1) such ha ˜
σ =σ and λµ =µ˜
. The e
exis ∈Zand N∈Nsuch ha pαn−nλ(xn)= pαn−nxn,∀n≥N
(Lemma 2.1). We ha e o each n≥N,µ[pαn−n(˜
(Xn)− Xn)] = 0, so
ha ( (an)− an)∈pnA.
Le us pu (an)=knan(Co olla y 3.2), hen we ha e pndi ides
kn− ,∀n≥N.Byusing again Co olla y 3.2, we can es ablish easily
ha (an)= an,∀n≥1. Le b∈Tqwi h q=p, and pu o(b)=qs.
Le us conside he ee g oup L=
n≥0
en. Le L0be he subg oup o L
gene a ed by {qse0}∪{qαnen−e0/n ≥1}.