Publ. Ma . 48 (2004), 397–408
RINGS WHOSE CLASS OF PROJECTIVE MODULES IS
SOCLE FINE
Abdelouahab Idelhadj, El Amin Kaidi, Dolo es Ma ´
ın
Ba que o and C´
andido Ma ´
ın Gonz´
alez
Abs ac
A class Co modules o e a uni a y ing is said o be socle ine
i whene e M, N ∈ C wi h Soc(M)∼
=Soc(N) hen M∼
=N. In
his wo k we cha ac e ize ce ain ypes o ings by equi ing a
sui able class o i s modules o be socle ine. Then we s udy socle
ine classes o quasi-injec i e, quasi-p ojec i e and quasicon inu-
ous modules which we apply o ind socle ine classes in special
ypes o noe he ian ings. We also ini ia e he s udy o hose ings
whose class o p ojec i e modules is socle ine.
1. In oduc ion
The no ion o socle ine class o modules has been p e iously used in
he algeb aic li e a u e wi hou an explici o mula ion. Thus in [5, The-
o em 9.3.7, p. 166] i is p o ed ha an algeb a Ais quasi-F obenius i and
only i each p incipal A-module has a simple socle and, o any wo non-
isomo phic p incipal A-modules P1and P2, we ha e Soc(P1)6∼
=Soc(P2).
Clea ly, his las asse ion is equi alen o he ac ha he class o p in-
cipal A-modules is socle ine. Also, he esul gi en in [5, Co olla y 9.4.3,
p. 171] could be s a ed by p oclaiming ha any class o indecomposable
modules o he same ini e leng h, o e a unise ial algeb a, is socle ine.
O he esul s in his ein appea in [15] whe e some socle ine classes a e
ound in he con ex o CEP- ings. Mo e ecen ly, Page and Zhou in [23,
Theo em 24, p. 2920] ind a se ies o equi alen condi ions cha ac e izing
he socle ine cha ac e o ce ain na u al classes. An explici o mula ion
o he no ion was gi en by A. Idelhadj and E. A. Kaidi in [11] whe e hey
2000 Ma hema ics Subjec Classi ica ion. P ima y: 13C13; Seconda y: 16D10.
Key wo ds. Socle ine class, p ojec i e module, QI-module, QP-module.
Suppo ed by he Spanish DGICYT wi h p ojec numbe s BFM 2001–2335 and BFM
2001–1886, by he Jun a de Andaluc´ıa p ojec s: FQM-336, FQM 0194, and ‘Es udio
anal´ı ico-algeb aico de Sis emas T iples y de Pa es en di e en es clases de es uc u as
no asocia i as’.
398 A. Idelhadj e al.
gi e a socle ine cha ac e iza ion o semia inian ings. Con inuing his
philosophy, a inian and noe he ian ings a e cha ac e ized in [16] by
socle ine classes. The in e es ing pape [12] con ains also cha ac e iza-
ions o V- ings and pseudo-F obenius ings. Complemen a y li e a u e
on socle and adical ine classes can be ound in [13], [17] and [10].
Th oughou his wo k he wo d ing will mean a uni a y (associa i e)
ing and modules a e unde s ood as uni a y (le ) modules. I Ris a ing
and Xa class o R-modules, we shall say ha Xis socle ine ( espec i ely
adical ine) whene e o any M, N ∈Xwe ha e Soc(M)∼
=Soc(N)
( espec i ely M/ Rad(M)∼
=N/ Rad(N) ) i and only i M∼
=N. We
shall deno e he injec i e hull o a module Mby E(M).
2. QI, QP and quasi-con inuous modules
Ou main e e ences o quasi-injec i e (QI), quasi-p ojec i e (QP)
and quasi-con inuous modules a e [22] and [1]. By in oducing QI
o QP modules in o he scene, we ob ain ano he cha ac e iza ion o
semisimple ings:
Theo em 2.1. Fo any ing R, he ollowing asse ions a e equi alen :
1) Ris semisimple.
2) The class o all QP modules is socle ine.
3) The class o all QI modules is socle ine.
4) The class Co all he ini e di ec sums o QP modules is socle ine.
P oo : I Ris semisimple he class o all R-modules is socle ine hence
1) implies 2), 3) and 4). Suppose 2). As Rand Soc(R) a e QP and
Soc(R) = Soc(Soc(R)) we ha e R∼
=Soc(R) implying ha Ris semisim-
ple. I 3) holds, since Soc(E(R)) = Soc(R) = Soc(Soc(R)) and bo h
o E(R) and Soc(R) a e QI, we ha e E(R)∼
=Soc(R) is semisimple and
so Ris also. Finally suppose 4). We shall p o e ha Ris semisimple
by showing ha he class o he QP R-modules is closed unde ini e
di ec sums. Le X1,...,Xnbe QP R-modules, and X=⊕iXibe an
elemen in C, le Y:= Soc(X) which is QP by i s semisimple cha ac e .
Since X, Y ∈ C and Soc(X) = Soc(Y), we conclude X∼
=Yhence Xis
semisimple and he e o e QP. Now he ac ha he class o QP modules
is closed unde ini e di ec sum implies ha Ris semisimple (see [7]
and [20]).
Le Cbe a socle ine class, Man elemen in Cand [M] he isomo phism
class o M. Then he class C ∪[M] is also socle ine. Mo eo e i {Mi}i∈I
is a collec ion o elemen s in C, he class o R-modules C ∪ (∪i∈I[Mi]) is
P ojec i e Modules A e Socle-Fine 399
socle ine. I is ob ious ha i C1and C2a e wo R-module socle ine
classes such ha ∀X∈ C1,∀Y∈ C2, Soc(X)6∼
=Soc(Y) hen C1∪ C2is
a socle ine class.
Theo em 2.2. Le Mbe a quasi-con inuous R-module and C he class
o i s di ec summands. Then Cis socle ine i and only i Soc(M)is
essen ial in M.
P oo : Le us suppose ha Cis socle ine. As E(M) = E(Soc(M)) ⊕W
wi h Soc(W) = 0 and Mis quasi-con inuous, by [22, Theo em 2.8, (4),
p. 20] we ha e ha M= (M∩E(Soc(M)))⊕(M∩W), whe e Soc(M) is
essen ial in M∩E(Soc(M)) and Soc(M)∩W= 0. Since M∩Wis a di ec
summand o Mwi h ze o socle and Cis socle ine, M∩W= 0 and Soc(M)
is essen ial in M. Le us suppose now ha Soc(M) is essen ial in M.
Le Nbe a di ec summand o M. Then Soc(N) is essen ial in Nhence
E(N) = E(Soc(N)). The e o e i N1, N2∈ C, wi h Soc(N1)∼
=Soc(N2)
we ha e ha E(N1)∼
=E(N2) and by [22, Theo em 2.31, p. 34], we ha e
N1∼
=N2.
Co olla y 2.1. Le Rbe a ing. Then Ris a V- ing i and only i he
class Co i s indecomposable QI modules wi h essen ial socle is socle
ine. The ing Ris a noe he ian V- ing i and only i he class Do i s
QI modules wi h essen ial socle is socle ine.
P oo : Le Rbe a V- ing and Mbe an indecomposable QI R-module.
Then E(M) is also indecomposable (see [22, Theo em 2.8, p. 20]). Le S
be a simple (hence injec i e) submodule o M. Then E(M) = Sand
so M=S. By he p e ious co olla y, he class Cis socle ine. Re-
cip ocally, i Cis socle ine ake a simple R-module S. Then E(S) is
indecomposable and he e o e S, E(S)∈C. Since Soc(S) = Soc(E(S))
we ha e S∼
=E(S) and so Sis injec i e. Thus Ris a V- ing.
Suppose now ha Ris a noe he ian V- ing. I is easy o p o e ha
he indecomposable QI modules a e semisimple hence hey o m a socle
ine class D. On he o he hand, i Dis socle ine, by he p o ed p e ious
pa o his co olla y Ris a V- ing. Nex we p o e ha any semisimple
module is injec i e: le M=⊕iSiwi h each Sia simple submodule.
Then M∈Dand as Soc(E(M)) = Soc(M) = Mwe deduce ha E(M)
has an essen ial socle; hen M, E(M)∈Dand Soc(M) = Soc(E(M))
hence M∼
=E(M) and so any semisimple module is injec i e. This
implies ha he class in [16, Theo em 3] is socle ine and he e o e Ris
noe he ian.
400 A. Idelhadj e al.
Co olla y 2.2. Le Dbe a noe he ian domain wi h K ull dimension 1
and Cany class o pai wise ela i ely injec i e o sion D-modules. Then
Cis socle ine.
P oo : By Theo em 2.2 i su ices o p o e ha any elemen M∈ C has
essen ial socle. Since Mis quasi-injec i e, each submodule o Mis essen-
ial in a summand o M(see [22, P oposi ion 2.1, p. 18]). Then M=T⊕
Wwhe e Soc(M) is essen ial in T(and Soc(W) = 0). As E(W) is a di-
ec sum o indecomposable injec i e D-modules, applying he co olla y
o [26, Theo em 2.32, p. 53] each injec i e indecomposable module is o
he o m E(D/P) wi h Pa p ime ideal o D. As Soc(W)=Soc(E(W))=0.
Then each indecomposable componen E(D/P) o E(W) has ze o socle
hence Pis no maximal, and so, since he K ull dimesion o Dis one,
P= 0. Consequen ly E(W) = ⊕i∈IQi, wi h Qi=Q(D) he ield o
ac ions o D. Then T(E(W)) = 0 implying T(W) = 0. Fu he mo e
M=T(M) = T(T)⊕T(W) = T(T)⊂T⊂M, hence M=T,W= 0
and Soc(M) is essen ial in M.
Co olla y 2.3. I Dis a Dedekind domain, he class o he indecom-
posable D-modules o he same ( ini e) leng h is socle ine.
P oo : This co olla y is a consequence o he s uc u e heo y o quasi-
injec i e modules o e Dedekind domains. We ecall ha , in his con ex ,
any QI module is ei he injec i e o a o sion D-module Msuch ha
o each nonze o p ime ideal P, he P-p ima y componen MPo Mis
a di ec sum o isomo phic modules each one o hem being isomo phic
o D/P n(n > 0) o o E(D/P) (see [9]). The indecomposable D-mod-
ules o ini e leng h a e o sion modules (see [21, Theo em 1, p. 49]),
by he s uc u e heo y o ini ely gene a ed modules o e a Dedekind
domain, hey ha e he o m D/P nwi h Pa p ime nonze o ideal and
nag eeing wi h he leng h o D/P n. As a consequence, hese D-modules
a e QI. Mo eo e he D-module (D/P n)⊕(D/Qn) wi h P6=Q, is also QI
hence D/P nand D/Qna e ela i ely injec i e [1, P oposi ion 2.2, p. 15].
Then applying Co olla y 2.2, he class o he indecomposable D-modules
o he same ini e leng h is socle ine and he co olla y is p o ed.
Le Dbe a Dedekind domain and Ta nonze o o sion D-module.
Then Thas a di ec summand isomo phic ei he o E(D/P) o o D/P nP
o some maximal ideal Po Dand nP∈N− {0}(see [18]). By using
his, i can be p o ed ha Mis a QP (quasi-p ojec i e) D-module i
and only i Mis ei he a p ojec i e module o a o sion D-module such
ha o each maximal ideal P, i s P-p ima y componen is a di ec sum
o modules all isomo phic o R/P np o some np∈N− {0}. I is easy o
P ojec i e Modules A e Socle-Fine 401
p o e (see [1, Exe cise 18, p. 24]) ha i Dis a Dedekind domain and Ma
ini ely gene a ed o sion D-module hen Mis QI, i and only i Mis QP,
i and only i M∼
=(D/P n1
1)m1⊕(D/P n2
2)m2⊕ · · · ⊕ (D/P nk
k)mk, whe e
P1, P2,...,Pka e di e en maximal ideals o D, and mi, ni∈N− {0}
o each i= 1,2,...,k.
P oposi ion 2.1. Le Dbe a Dedekind domain. I Cis a class o QP
D-modules wi h nonze o socle and all o hem wi h isomo phic adical,
hen Cis socle ine. In pa icula any class o QI ini ely gene a ed
o sion modules wi h isomo phic adicals is socle ine.
P oo : We conside Mand N wo elemen s o he class Cwi h isomo phic
socles. By he p e ious pa ag aph
M∼
=M
i∈I
(D/P mi
i)(Ai),
N∼
=M
j∈J
(D/Qnj
j)(Bj)
whe e Piand Qja e maximal, and Ai,Bja e nonemp y se s. By he ac
ha he socles a e isomo phic we ha e he exis ence o a bijec ion σ:I→J
such ha |Ai|=|Bσ(i)|,Pi=Qσ(i) o all i∈I. On he o he hand, a e
a sui able eo de ing, we can w i e N∼
=Li∈I(D/P ni
i)(Ai) he e o e
Rad(M)∼
=M
i∈I
(D/P mi−1
i)(Ai),
Rad(N)∼
=M
i∈I
(D/P ni−1
i)(Ai)
and as hei adicals a e isomo phic hei p ima y componen s a e iso-
mo phic also. Thus (D/P mi−1
i)(Ai)∼
=(D/P ni−1
i)(Ai). Fu he mo e hei
annihila o s ag ee, ha is o say, Pmi−1
i=Pni−1
iimpliying mi=ni o
all i.
3. Rings whose class o p ojec i e modules is socle ine
I has been men ioned in he in oduc ion, ha he ings whose class
o injec i e modules is socle ine a e p ecisely he semia inian ings. I
is he e o e na u al o pose he ques ion on he ings wi h socle ine class
o p ojec i e modules. One i s app oach o he p oblem is gi en by he
nex heo em.
402 A. Idelhadj e al.
Theo em 3.1. Le Rbe a ing and F he class o he ee R-modules.
Then he ollowing asse ions a e equi alen :
1) Fis socle ine.
2) Ris an IBN ing wi h Soc(R)6= 0 and some homogeneous compo-
nen in Soc(R)has ini e leng h.
P oo : Suppose i s ha Fis socle ine. I Soc(R) = 0 o no componen
has a ini e leng h one checks immedia ely ha Soc(R)∼
=Soc(RN) which
ake us o he con adic ion R∼
=RN. Nex we p o e ha Ris IBN. I
Rn∼
=Rm o n, m ∈N hen Soc(Rn)∼
=Soc(Rm). Since Soc(R) =
Sk0
i0⊕(⊕i6=i0S(Ii)
i) wi h k0∈N∗, and Si0,Sihomogeneous componen s,
his gi es Sk0n
i0⊕(⊕i6=i0S(Ji)
i)∼
=Sk0m
i0⊕(⊕i6=i0S(Hi)
i) whence k0n=
k0mand so n=m, as equi ed. Le us p o e 2) ⇒1). Conside wo
ee modules R(I),R(J)wi h isomo phic socles and Soc(R) = Sk0
i0⊕
(⊕i6=i0S(Ii)
i) wi h k0∈N∗, and Si0,Si he homogeneous componen s.
Then:
Soc(R(I)) = S(X)
i0⊕(⊕i6=i0S(I×Ii)
i)
Soc(R(J)) = S(Y)
i0⊕(⊕i6=i0S(J×Ii)
i)
whe e X={1,...,k0} × I,Y={1,...,k0} × J. F om he hypo hesis
ha he socles a e isomo phic one ge s |X|=|Y|hence |I|=|J|.
We ecall ha a ing Ais le pseudo-F obenius (a le PF ing) i A
is an injec i e cogene a o . Le PF ings a e cha ac e ized by he nex
heo em:
Theo em 3.2. Le Abe a ing. Then he ollowing s a emen s a e
equi alen :
1) Ais a le PF ing.
2) The class o p ojec i e A-modules is socle ine and Ais a le co-
gene a o .
P oo : I Ais a le PF ing hen by de ini ion and [3] and [4] he second
asse ion holds. Suppose now ha Ais a le cogene a o wi h i s class o
p ojec i e modules being socle ine. Then Soc(A)6= 0 since i Soc(A) =
0 = Soc(0) hen A= 0. Thus Soc(A) = ⊕i∈ISi=⊕i∈ISoc(E(Si)).
As Ais a cogene a o hen each E(Si) embeds in Aand E(Si) is a di ec
ac o o A. Hence E(Si) is p ojec i e. We ha e hen ha ⊕i∈IE(Si)
is a p ojec i e A-module. Then since Soc(A) = Soc(⊕i∈IE(Si)) we ge
A∼
=⊕i∈IE(Si). As Ais o ini e ype A∼
=⊕n
i=1E(Si) o some n∈N,
whence Ais injec i e and as a consequence Ais a le PF ing.
P ojec i e Modules A e Socle-Fine 403
Theo em 3.3. Le Abe a ing, hen he ollowing p ope ies a e equi -
alen :
1) Ais a QF ing.
2) The class Do p ojec i e o injec i e modules is socle ine.
P oo : I Ais QF, he class Dis jus he class o injec i e modules (and
also he class o p ojec i e ones by [19, Theo em 13.6.1, p. 352]). Since A
is a inian, his class is socle ine by [16, Theo em 2]. Nex we p o e ha
i Dis socle ine, hen Ais a QF ing. Take Pa p ojec i e module, hen
we ha e Soc(P) = Soc(E(P)) and P, E(P)∈ D. Consequen ly P∼
=E(P)
and so Pis injec i e. By [19, Theo em 13.6.1, p. 352], Ais QF.
P oposi ion 3.1. Le Abe a cogene a o ing. Then he ollowing as-
se ions a e equi alen :
1) Ais a le QF3 ing.
2) The class o p ojec i e A-modules is socle ine.
P oo : I Ais a cogene a o le QF3 ing hen Ais a PF- ing by [25,
p. 55]. Now by Theo em 3.2 he class o p ojec i e A-modules is socle
ine. The o he implica ion is i ial applying Theo em 3.2.
P oposi ion 3.2. Le Abe a le QF3 ing. Then he ollowing asse -
ions a e equi alen :
1) Ais a QF- ing.
2) The class o p ojec i e A-modules is socle ine.
P oo : I Ais a QF- ing hen he class o p ojec i e A-modules ag ees
wi h he class o injec i e modules and as Ais an a inian ing, his class
is socle ine (see [16, Theo em 2]). Suppose ha he class o p ojec i e
A-modules is socle ine. Le Pbe a p ojec i e A-module and E(P) i s
injec i e hull. By [8, Co olla y II.6, p. 58], bo h Pand E(P) a e p o-
jec i e. Since Soc(P) = Soc(E(P)) his implies P∼
=E(P) whence Pis
injec i e and Ais a QF- ing.
4. Semipe ec ings
Le Abe a ing and Ji s Jacobson adical. We ecall ha Ais
semipe ec i A/J is semisimple and any idempo en o A/J is o he
o m e+Jwi h ean idempo en o A. I is well known (see o ins ance [2,
P oposi ion 27.10, p. 306]) ha i Ais semipe ec , each comple e amily
o p imi i e idempo en s con ains a basic amily e1,...,emo A, and all
he basic amilies o Aha e he same ca dinali y which is called he ca-
paci y o A. The A-modules Ae1,...,Aem o m a comple e i edundan
404 A. Idelhadj e al.
amily o ep esen a i es o p ojec i e indecomposable A-modules. The
A-modules Ae1/Je1,...,Aem/Jem o m a comple e i edundan amily
o ep esen a i es o simple A-modules. We shall use he no a ion Si=
Aei/Jei o all i. I Pis a p ojec i e A-module, hen Phas an essen ially
unique decomposi ion o he ype P= (Aei)(I1)⊕ · · · ⊕ (Aem)(Im). A
ing Ais called ini ely embedded i and only i i has an essen ial and
ini ely gene a ed socle. Any a inian ing is ini ely embedded bu he e
a e ings which a e ini ely embedded (and e en local) bu nona inian
(see [24]).
Theo em 4.1. Le Abe a ini ely embedded semipe ec ing. The ol-
lowing asse ions a e equi alen :
1) The class o p ojec i e A-modules is socle ine.
2) Acon ains all i s ypes o simple A-modules, and any p ojec i e
indecomposable module has a homogeneous socle.
P oo : Conside a ini ely embedded semipe ec ing Awi h capaci y m
and a basic amily e1,...,emo A. Suppose ha he class o p ojec i e
A-modules is socle ine. I m= 1 hen asse ion 2) ollows om he ac
ha Soc(A)6= 0. Nex we ake m≥2, and suppose (a e a sui able
eo de ing i necessa y) ha Soc(A)∼
=Sα1
1⊕ · · · ⊕ Sα
wi h αi6= 0 o
i∈ {1,..., }, and < m. Then he e is a ≤ such ha S1⊕ · · · ⊕ S
can be embedded in Aei1⊕ · · · ⊕ Aei . Consequen ly Soc(Aei1⊕ · · · ⊕
Aei ) = Sβ1
1⊕ · · · ⊕ Sβ
wi h βi6= 0 o all i∈ {1,..., }. The p ojec i e
A-modules A(N)and (Aei1⊕ · · · ⊕ Aei )(N)ha e isomo phic socles hence
we ha e an isomo phism
Ae(N)
1⊕ · · · ⊕ Ae(N)
m∼
=Ae(N)
i1⊕ · · · ⊕ Ae(N)
i .
Since < m he e is a j∈ {1,...,m}such ha j6∈ {i1,...,i }, bu on
he o he hand, he heo em on he uniqueness o he decomposi ion o
p ojec i e A-modules implies ha Aej∼
=Aeik o some ik∈ {i1,...,i }
which is con adic o y. Nex we p o e ha Soc(Aei) is homogeneous o
each i. Suppose ha Soc(Aek) = Sα1
1⊕ · · · ⊕ Sαq
qwi h q≥2 and αi6= 0
o all i. Then he e exis Aei1,...,Aeinpai wise di e en and dis inc
om Aeksuch ha n≤m−q, and Sq+1 ⊕ · · · ⊕ Smcan be embedded
in Aei1⊕ · · · ⊕ Aein. Consequen ly Soc(Aek⊕Aei1⊕ · · · ⊕ Aein) =
Sβ1
1⊕ · · · ⊕ Sβm
mwi h each βi6= 0. As (Aek⊕Aei1⊕ · · · ⊕ Aein)(N)
is p ojec i e and i s socle is isomo phic o he socle o he p ojec i e
A-module A(N)we ha e an isomo phism
Ae(N)
k⊕Ae(N)
i1⊕ · · · ⊕ Ae(N)
in
∼
=Ae(N)
1⊕ · · · ⊕ Ae(N)
m
P ojec i e Modules A e Socle-Fine 405
which implies ha n+ 1 = mby he p e iously men ioned uniqueness
heo em. Since we had n≤m−qand q≥2, hen n+ 1 ≤m−q+ 1 ≤
m−1, a con adic ion. Suppose now ha 2) holds, and deno e by Aeσ(i)
he unique di ec summand o Acon aining o Si. I is clea ha i7→ σ(i)
is a pe mu a ion o {1,...,m}and Soc(Aeσ(i))∼
=Sni
iwi h ni6= 0.
Take Pand Q wo p ojec i e A-modules wi h P=⊕m
j=1Ae(Ij)
σ(j)and
Q=⊕m
j=1Ae(Hj)
σ(j). I Soc(P)∼
=Soc(Q) we ha e ha ⊕j(Snj
j)(Ij)∼
=
⊕j(Snj
j)(Hj)and acco ding o he uniqueness o he homogeneous com-
ponen s o semisimple modules we ha e (Snj
j)(Ij)∼
=(Snj
j)(Hj) o all j.
The uniqueness o he decomposi ion o a semisimple module as a di-
ec sum o simple ones implies he coincidence o ca dinals: |Ij|=|Hj|
whence P∼
=Q.
Co olla y 4.1. Le Abe ing, o some o he ollowing ypes:
1) A ini ely embedded local ing.
2) A p ima y ing ( ha is Ais a inian and A/ Rad(A)is simple).
3) A commu a i e a inian ing.
Then he class o p ojec i e A-modules is socle ine.
P oo : Suppose ha Ais as in he i s possibili y. Since any local
ing is semipe ec and all he simple A-modules a e isomo phic, om
Soc(A)⊂Awe conclude ha Acon ains i s unique ype o simple
A-module. As any p ojec i e A-module is ee and Soc(A) is homo-
geneous we ha e ha any p ojec i e indecomposable module has a ho-
mogeneous socle. In he second case, ake in o accoun ha he simple
modules o e a simple a inian ing a e isomo phic. I is easy o p o e
ha he simple A-modules o he o m Aei/Rad(A)eia e also simple as
A/ Rad(A)-modules. Thus hey a e isomo phic as A/ Rad(A)-modules
and also as A-modules. In his way he e is only one isomo phism class
o simple A-modules and he condi ions in i em 2 o Theo em 4.1 a e
sa is ied. Finally, i Ais commu a i e and a inian, i spli s in o a ini e
di ec sum o local a inian ings. The class o p ojec i e modules o any
o hese summands is socle ine (as p o ed in he i s i em), and om
his i is easy o de i e ha he class o p ojec i e A-modules is socle
ine.
In he noncommu a i e case we do no ha e in gene al his p ope y.
Take o ins ance a ield Kand A he a inian ing o iangula ma ices
de ined by:
A=K0
K K.
