u n co rr ec te d p ro o f
O R I G I N A L P A P E R
Rings Characterized via Some Classes of Almost-Injective Modules
Truong Cong Quynh1·Adel Abyzov2·Phan Dan3·Le Van Thuyet4
Received: 27 May 2020 / Revised: 27 May 2020 / Accepted: 31 August 2020
© Iranian Mathematical Society 2020
Abstract
1
In this paper, we study rings with the property that every cyclic module is almost-
2
injective (CAI). It is shown that R is an Artinian serial ring with J(R)2 = 0 if and
3
only ifR is a right CAI-ring with the finitely generated right socle (or I-finite) if and 1
4
only if every semisimple right R-module is almost injective, RR is almost injective
5
and has finitely generated right socle. Especially, R is a two-sisded CAI-ring if and
6
only if every (right and left) R-module is almost injective. From this, we have the
7
decomposition of a CAI-ring via an SV-ring for which Loewy (R)≤2 and an Artinian
8
serial ring whose squared Jacobson radical vanishes. We also characterize a Noetherian
9
right almost V-ring via the ring for which every semisimple rightR-module is almost
10
injective.
11
Keywords Almost-injective module·Almost V-ring·V-ring·CAI-ring
12
Mathematics Subject Classification 16D50·16D70·16D80
13
Communicated by Mohammad-Taghi Dibaei.
B
Le Van Thuyet[email protected] Truong Cong Quynh [email protected] Adel Abyzov
[email protected] Phan Dan
1 Department of Mathematics, The University of Danang-University of Science and Education, DaNang City, Vietnam
2 Department of Algebra and Mathematical Logic, Kazan (Volga Region) Federal University, Kazan 420008, Russia
3 Hong Bang International University, Ho Chi Minh City, Vietnam
4 Department of Mathematics, College of Education, Hue University, Hue, Vietnam
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1 Introduction
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Throughout this paper, all rings Rare associative with unit and all modules are right
15
unital. Let M and N be right R-modules. The module M is said to be almost N-
16
injective(oralmost injective respect to N) if, for every submodule N1of N and for
17
every homomorphism f : N1 → M,either there is a homomorphismg : N → M
18
such that f =g◦ι, i.e., the diagram (1) commutes, or there is a nonzero idempotent
19
π ∈ End(N)and a homomorphismh : M → π(N)such thath ◦ f = π ◦ι, i.e.,
20
the diagram (2) commutes, whereι : N1 → N is the embedding of N1 into N. The
21
module M is said to bealmost injectiveif it is almost injective with respect to every
22
right R-module.
23
(1)
0 N1 N
M
✲ ι✲
❄
f
♣
♣
♣
♣
♣
♣
✠♣
g
0 N1 N
M π(N)
✲ ι✲
❄
f
❄
π h✲
(2)
24
This concept was defined by Baba in many years ago, however, many related results
25
were obtainned in recent years, for examples, see [1–8], ... Of course, injective⇒
26
almost injective, but the converse isn’t true, in general. It is proved that a ring R is
27
semisimple if and only if every right (left) R-module is injective and then a well-
28
known result of Osofsky said that it is equivalent to every cyclic right (left)R-module
29
is injective. In [4], the authors consider the structure of a ring R over which every
30
module is almost injective. It is natural to ask how is the structure of a ringRfor which
31
every cyclic module is almost injective. We continue prove that the class of rings whose
32
all cyclic right R-modules are almost injective contains the class of Artinian serian
33
rings with squared Jacobson radical vanishes. So Theorem 1 and it’s Corollaries from
34
[4] are followed from our result, i.e., in cases of if Soc (RR) is finitely generated
35
(or R is semiperfect, or RR is extending, or R is of finite reduced rank), then two
36
above classes and the class of the rings whose all rightR-modules are almost injective
37
coincide. Especially, a ringRis two-sided CAI if and only if every (right and left) R-
38
module is almost injective. From this result, we have the decomposition of a CAI-ring
39
via an SV-ring for which Loewy(R)≤ 2 and an Artinian serial ring whose squared
40
Jacobson radical vanishes.
41
Recall that Ris a rightV-ringif every simple right R-module is injective. In [3],
42
the authors consider a generalization of a V-ring, that is almost V-ring, i.e., if every
43
simple right R-module is almost injective. A module M is called simple-extending
44
(semisimple-extending, resp.) if the complement of any simple (semisimple, resp.)
45
submodule of M is a direct summand of M. Now we write the class 1 stands for
46
all rings R for which every simple module is almost injective, i.e., R is an almost
47
V-ring, theclass 2stands for all ringsRfor which every semisimple module is almost
48
injective, theclass 3stands for all ringsRfor which every module is simple-extending.
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In [3], the authors proved that the class 1 and class 3 coincides (see [3], Theorem
50
2.9). It is also proved that the intersection of the class 1 and the class of all right
51
Noetherian rings is equal to the class 2 (see [5], Theorem 2.4). Our aim is to consider
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the weaker conditions of Noetherian, that are having finite Goldie dimesion or finitely
53
generated right socle together the class 1 will be replaced by class 2 and we also
54
obtain a characterization of a right Noetherian right almost V-ring. From this, we give
55
back some characterizations of an Artinian serial ring with squared Jacobson radical
56
vanishes via class 2.
57
For a submoduleNofM, we useN ≤ M(N < M) to mean that Nis a submodule
58
ofM(respectively, proper submodule), and we writeN ≤e Mto indicate thatN is an
59
essential submodule of M. A module is called aCS-module, orextending, provided
60
every complement submodule is a direct summand. A module is calleduniformif the
61
intersection of any two nonzero submodules is nonzero. A ringRis calledI-finiteif it
62
contains no infinite orthogonal family of idempotents. LetMbe an arbitrary module.
63
Recall that Z(M) = {m ∈ M|m I = 0 for some I ≤e RR} is called the singular
64
submoduleof M, and if Z(M) = M (Z(M) = 0, resp.), then M is calledsingular
65
(nonsingular. resp.) (see [9]). TheGoldie torsion(orsecond singular)submoduleof
66
Mdenoted byZ2(M)satisfies Z(M/Z(M)) = Z2(M)/Z(M). The (Goldie)reduced
67
rankof M is the uniform dimension of M/Z2(M). Every module of finite uniform
68
dimension is of finite reduced rank. Let M,N be arbitrary modules. M is called
69
essentially N-injectiveif for every embeddingι: A→ N and every homomorphism
70
f : A → M such that Kerf ≤e A, there exists a homomorphismg : N → M such
71
that ι◦g = f. The module M is said to be essentially injectiveif it is essentially
72
N-injective with each N ∈ Mod-R. Moreover, Ris a rightSC-ringif every singular
73
R-module is continuous.M is called auniserial module, if the set of submodules of
74
M is linear ordered. A ring R is called semiperfect in case R/J(R)is semisimple
75
and idempotents lift moduloJ(R). It is equivalent to every its finitely generated right
76
(left)R-module has a projective cover. A ring R is called a rightperfect ringin case
77
R/J(R)is semisimple and J(R)is right T-nilpotent. It is equivalent to every its right
78
R-module has a projective cover.
79
By theLoewy seriesof a moduleMR we mean the ascending chain
80
0≤Soc1(M)=Soc(M)≤ · · · ≤Socα(M)≤Socα+1(M)≤ . . . ,
81
where
82
Socα(M)/Socα−1(M)=Soc(M/Socα−1(M))
83
for every nonlimit ordinalαand
84
Socα(M)=
β<α
Socβ(M)
85
for every limit ordinal α. Denote by L(M) the submodule of the form Socξ(M),
86
whereξ stands for the least ordinal for which Socξ(M)=Socξ+1(M).A module M
87
is semiartinian if and only ifM = L(M).In this case,ξis called theLoewy lengthof
88
the moduleMand is denoted by Loewy(M). A ringRis said to beright semiartinianif
89
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the moduleRRis semiartinian. In this case, every nonzero (principal) rightR-module
90
has a nonzero socle and a ringRis right perfect if and only if it is left semiartinian and
91
I-finite. The class of right semiartinian right V-rings, which we callright SV-rings.
92
A ring R is called right nonsingular if Z(RR) = 0, right serial if RR is a direct
93
sum of uniserial modules. In this paper, we denote by Rad(M), Soc(M), E(M), and
94
length(M)the Jacobson radical, the socle, the injective hull and the composition length
95
ofM, respectively. The full subcategory of Mod-Rwhose objects are all R-modules
96
subgenerated byMis denoted byσ[M].
97
Left-sided for these above notations are defined similarly. All terms such as
98
“artinian”, “serial”, ... when applied to a ring will apply all both sided. For any terms
99
not defined here the reader is referred to [9–13].
100
2 On Rings with Cyclic Almost-Injective Modules
101
Firstly, we include the following known result related to finite decomposition of
102
almost-injective modules for the sake of completeness.
103
Lemma 2.1 [8, Lemma 1.14] Let N,V1,V2, . . . ,Vn be a family of modules over a
104
ring R. Then M = ni=1Vi is almost N -injective if and only if every Vi is almost
105
N -injective.
106
The second author gave the following problem in [1]: describe the rings over which
107
every cyclic right R-module is almost-injective. In this section, we will study on
108
this problem and give some characterizations of rings for which every cyclic right
109
R-module is almost-injective.
110
Definition 2.2 A ringRis calledright CAI, if every cyclic right R-module is almost-
111
injective. IfRis a right and left CAI-ring, then Ris called a CAI-ring.
112
Example 2.3 (1) Every semisimple ring is CAI.
113
(2) LetF be a field. Then, the ring R= F F
0 F is a right CAI-ring.
114
Firstly, we give the following key lemma:
115
Lemma 2.4 Let R be a right CAI-ring. If M is a right R-module, then M/A is a
116
semisimple module for every essential submodule A of M.
117
Proof Let A be an essential submodule of M. We show that M/A is a semisimple
118
module. By [11, Corollary 7.14], it is necessary to prove that every cyclic right R-
119
module in the category σ[M/A] is M/A-injective. In fact, let N be a cyclic right
120
R-module (in the categoryσ[M/A]) and f : A/A → N be a homomorphism from
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an arbitrary submodule A/A of M/A to N. We show that f is extended to M/A.
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Callπ1 : A → A/A, π2 : M → M/A the natural projections andι1 : A → M,
123
ι2: A/A→ M/Athe inclusions. We consider the homomorphism f ◦π1 : A → N.
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We show that f ◦π1is extended to M. Otherwise, since N is almost-injective, there
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exist an idempotent π of End(M)and a homomorphismh : N → π(M)such that
126
π◦ι1 =h◦(f ◦π1).
127
A M
N π(M)
ι1✲
❄
f◦π1
❄
π h✲
128
Then, we have
129
π(A)=(π ◦ι1)(A)=(h◦ f)(π1(A))=0.
130
It means that A ≤ Ker(π ) = (1−π )(M), and so (1−π )(M) is essential in M.
131
This gives a contradiction. Thus, there is a homomorphism g : M → N such that
132
g◦ι1 = f ◦π1.
133
0 A M
N
✲ ι✲1
❄
f◦π1
♣
♣
♣
♣
♣
♣
✠♣
134 g
We have
135
g(A)=(g◦ι1)(A)=(f ◦π1)(A)=0
136
It shows that there is a homomorphismg : M/A → N such thatg =g ◦π2. From
137
this gives
138
f ◦π1=g◦ι1=(g ◦π2)◦ι1=g ◦(π2◦ι1)=g ◦(ι2◦π1)
139
It follows that f =g ◦ι2.Thus, N is M/A-injective.
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Corollary 2.5 Every right CAI-ring is a right SC-ring.
141
From Lemma2.4and [14], we have the following fact:
142
Fact 2.6 If R is a right CAI-ring, then
143
1. J(R)≤Soc(RR).
144
2. J(R)2 =0.
145
3. R/Soc(RR)is a right Noetherian ring.
146
Theorem 2.7 The following statements are equivalent for a ring R:
147
1. R is an Artinian serial ring with J(R)2 =0.
148
2. R is a right CAI-ring and R/J(R)is I-finite.
149
3. R is a I-finite right CAI-ring.
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4. R is a right CAI-ring with the finitely generated right socle.
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Proof (1)⇒(2)⇒(3)are obvious.
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(3) ⇒ (4)Suppose that R is a I-finite right CAI-ring. Then there exist primitive
153
idempotents e1,e2. . . ,en such that 1 = e1 +e2 + · · · +en. Note that all eiR are
154
indecomposable modules. Since R is a right CAI-ring, by [7, Lemma 3.1, Theorem
155
3.5], theneiR is uniform and End(eiR)is local for all i ∈ {1,2, . . . ,n}. It follows
156
that Ris a semiperfect ring. We deduce, from Fact 2.6, that R is a semiprimary ring
157
withJ(R)2 =0. Moreover, inasmuch aseiRis uniform which implies that Soc(eiR)
158
is simple for alli ∈ {1,2, . . . ,n}. Thus, Soc(RR)is finitely generated.
159
(4)⇒(1)Assume thatRis a right CAI-ring with the finitely generated right socle.
160
Then,Ris a right Noetherian ring by Fact2.6. We can writeR=e1R⊕e2R⊕· · ·⊕enR,
161
wheree1,e2. . . ,en are primitive idempotents such that 1= e1+e2+ · · · +en and
162
all right idealseiRare uniform. By the proof of(3)⇒ (4), R is a semiprimary ring
163
withJ(R)2 =0. We deduce that Ris a right Artinian ring. Note that(R⊕ R)R is an
164
extending right R-module by [7, Remark 3.2]. It follows that E(RR)is a projective
165
right R-module by [15, Theorem 3.3].
166
Next, we show thateiR is either simple or injective with the length of two. In
167
fact, for any nonzero submoduleU ofeiR, then eiR/U is a semisimple module by
168
Lemma2.4. Moreover,eiR/U is an indecomposable module. We deduce thateiRis
169
either simple or length of two. On the other hand, we have that E(eiR)is a uniform
170
projective module and obtain that E(eiR) ∼= ejR for some j ∈ {1,2, . . . ,n}. Now,
171
we assume thatekR is the module with length of two withk ∈ {1,2, . . . ,n}. Then
172
E(ekR) is indecomposable and projective. Therefore length (E(ekR)) ≤ 2, and so
173
E(ekR)=ekR,i.e.,ekRis injective. Thus,Ris an Artinian serial ring withJ(R)2=0
174
by [11, 13.5].
175
Corollary 2.8 The following statements are equivalent for a ring R.
176
1. R is an Artinian serial ring with J(R)2 =0.
177
2. R is a right CAI-ring with Soc(RR)/J(R)is finitely generated.
178
Example 2.9 Consider the ring R consisting of all eventually constant sequences of
179
elements fromF2. Clearly, R is a CAI-ring and Soc(R)is not finitely generated.
180
Lemma 2.10 If R is a right CAI-ring, then
181
1. R/Soc(RR)is semisimple.
182
2. R is a right semi-Artinian ring.
183
Proof (1) Assume that Ris a right CAI-ring. One can check that R/Soc(RR)is also
184
a right CAI-ring. From Fact2.6and Theorem2.7gives thatR/Soc(RR)is a right
185
Artinian ring. Note that R/Soc(RR)is a right V-ring by [3, Proposition 2.3]. We
186
deduce that R/Soc(RR)is semisimple.
187
(2) is followed from (1).
188
Proposition 2.11 Let R be a right CAI-ring. Then the followings hold:
189
1. Every direct sum of uniform right R-modules is extending.
190
2. Every uniform right R-module has length at most 2.
191
3. RR =(⊕i∈ILi)⊕N , where Li is a local injective module of length two for every
192
i ∈ I , J(N)=0andEnd(N)is a right SV -ring.
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Proof (1) From Lemma2.10, R is a right semiartinian ring. By [11, 13.1], we need
194
to prove that H1⊕ H2 is an extending module for any uniform modules H1and
195
H2. In fact, let H1 and H2 are uniform right R-module. Since H1 and H2 are
196
uniform with essential socles, Soc(H1⊕H2)is finitely generated and essential in
197
H1⊕H2. Inasmuch asRis a right CAI-ring, we have every simple rightR-module
198
is almost-injective, and so H1 ⊕ H2 is extending by [3, Theorem 2.9, Corollary
199
2.13.].
200
(2) is followed by (1) and [11, 13.1].
201
(3) By Zorn’s Lemma, there is a maximal independent set of submodules{Li}i∈I of
202
RRsuch thatLi is a local injective module of length two for everyi ∈ I.Since by
203
Fact2.6(3), R/Soc(RR)is a right Noetherian ring, then I is a finite set. Then, we
204
have a decompositionRR =(⊕i∈ILi)⊕N for some right ideal Nof R. Suppose
205
thatJ(N)=0. From Lemma2.10(2) gives J(N)containing a simple submodule
206
S.Let N0 be a complement of the submoduleS in the module N. It follows that
207
N/N0is a uniform nonsimple module whose socle is isomorphic to the moduleS.
208
Thus, it follows from (1) and [3, Theorem 3.1] thatN/N0is a projective module
209
and length of N/N0 is equal to two. Hence N = N0 ⊕ L, where L is a local
210
injective module of length two, which contradicts the choice of the set {Li}i∈I.
211
We deduce that J(N)=0.One can check that the moduleNcan be considered as
212
a projective R/J(R)-module. By [3, Proposition 2.3] and Lemma2.10, we have
213
R/J(R)is a right SV-ring. It follows from [16, Theorem 2.9] that End(N)is a
214
rightSV-ring.
215 216
For two-sided CAI-rings, we have:
217
Theorem 2.12 The following statements are equivalent for a ring R:
218
1. Every R-module is almost injective.
219
2. Every finitely generated R-module is almost injective.
220
3. R is a C A I -ring.
221
4. R is a direct product of an SV -ring for which Loewy(R) ≤ 2 and an Artinian
222
serial ring whose squared Jacobson radical vanishes.
223
Proof (1)⇒(2)⇒(3)are obvious.
224
(3)⇒ (4)By Proposition2.11, there exists an idempotente ∈ Rsuch that RR =
225
e R⊕(1−e)R, where e R = ⊕i∈ILi, Li is a local injective module of length two
226
for everyi ∈ I, J((1−e)R) =0 and(1−e)R(1−e)is a right SV-ring. One can
227
check that Hom(e R, (1−e)R) = 0 and J(R) = J(⊕i∈ILi). Then e R(1−e)is a
228
submodule of RR ande R(1−e) ≤ J(R). It follows, from the left-sided analogue
229
of Proposition2.11(3), that there exists a set of orthogonal idempotents{f1, . . . , fn}
230
such thate R(1−e) = J(R f1 ⊕ · · · ⊕ R fn)and R fi is a local injective module of
231
length two for every 1≤ i ≤ n.Consider the two-sided Peirce decomposition of the
232
ringRcorresponding to the decomposition 1=e+(1−e)
233
R = e Re eR(1−e) 0 (1−e)R(1−e) .
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Then for every 1≤i ≤nthe following equalities hold
235
fi = erie emi(1−e) 0 (1−e)si(1−e) ,
236
(erie)2 = erie, ((1−e)si(1−e))2 =(1−e)si(1−e)
237
and
238
emi(1−e)=eriemi(1−e)+emi(1−e)si(1−e).
239
LetS := (1−e)R(1−e)andgi := (1−e)si(1−e)for every 1 ≤ i ≤ n.Fix an
240
arbitrary index 1≤i ≤n.We have that
241
J(R)fi = e J(R)e e R(1−e)
0 0
erie emi(1−e)
0 gi ≤ 0e R(1−e)
0 0
242
and obtaine J(R)erie=0.On the other hand, for every j ∈ J(R)andm ∈e R(1−e)
243
we have
244
ej e em(1−e)
0 0
erie emi(1−e)
0 gi
= 0ej emi(1−e)+emgi
0 0
= 0ej e(eriemi(1−e)+emigi)+emgi
0 0
= 0e(j emi +m)gi
0 0 .
245
We deduce that J(R)fi ≤ 0e Rgi
0 0 .Since J(R)fi =0,then gi =0.Inasmuch
246
as the idempotent fi+J(R)∈ R/J(R)is primitive andJ(R)2 =0 we haveerie=0
247
ande J(R)e R(1−e)=0.Consequently,
248
0e R(1−e)
0 0 =
n
i=1
J(R)fi =
n
i=1
0 e R(1−e)gi
0 0 .
249
It means thate R(1−e) = ⊕in=1e R(1−e)gi ande R(1−e)(1− ni=1gi) = 0.If,
250
for some primitive idempotentg0of the ringS, the conditiong0S ∼=giSholds, where
251
1≤i ≤n, then it can readily be seen that Mg0 =0.Thus the right ideals
252
⊕n
i=1giSand (1−e)−
n
i=1
gi S
253
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of S do not contain isomorphic to simple S-submodules. Since S is a semiartinian
254
regular ring, theng = ni=1giis a central idempotent ofSand the ringRis isomorphic
255
to the direct product of the regular ring(1−e−g)Sand the ring
256
R = e Re e R(1−e)
0 g R .
257
Inasmuch as e R = e Re + e R(1 −e) is a module of finite length and for every
258
1≤i ≤n, the idempotentgi ∈ (1−e)R(1−e)is primitive, we obtain that the ring
259
R is Artinian. Thus the ring R is Artinian serial and J(R)2 = 0 by Theorem2.7.
260
From Proposition 2.11, we have(1−e−g)S is an SV-ring. Thus, the ring R is a
261
direct product of anS V-ring for which Loewy(R) ≤ 2 and an Artinian serial ring
262
whose squared Jacobson radical vanishes.
263
(4)⇒(1)is followed by Theorem2.7and [4, Proposition 2.6].
264 265
Theorem 2.13 The following statements are equivalent for a ring R:
266
1. R is a right hereditary C A I -ring.
267
2. R is a right nonsingular C AI -ring.
268
3. R is a direct product of an SV -ring for which Loewy(R)≤ 2and a finite direct
269
product of rings of the following form:
270
Mn1(T)Mn1×n2(T) 0 Mn2(T) ,
271
where T is a skew-field.
272
Proof (1)⇒(2)is obvious.
273
(2)⇒(3)is followed by Theorem2.12and [17, Theorem 8.11].
274
(3)⇒(1)is followed by [18, Proposition 9.6].
275 276
Corollary 2.14 Any I-finite right nonsingular right CAI-ring R is isomorphic to a finite
277
direct product of rings of the following form:
278
Mn1(T)Mn1×n2(T) 0 Mn2(T) ,
279
where T is a skew-field.
280
For two-sided CAI-rings, we obtain the important result, that is, they are also the
281
rings for which every (right and left)R-module is almost injective. So, it is natural to
282
ask the following question:
283
Question.Does the class of rings whose all rightR-modules are almost-injective and
284
class of all right CAI-rings coincide?
285
It is well-known that if M a non-singular indecomposable almost-injective right
286
R-module, then End(M)is an integral domain and every nonzero endomorphism of
287
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Mis a monomorphism. Moreover, ifM is a cyclic module over a right Artinian ring,
288
then End(M)is a skew-field. The following result is obvious.
289
Lemma 2.15 Let R be a right Artinian ring and e be a primitive idempotent of R. If
290
e R is a non-singular almost-injective right R-module, then e Re is a skew-field.
291
Lemma 2.16 Let R be a I-finite right nonsingular right CAI-ring and e, e be any two
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primitive idempotents in R with D=e Re and D =e Re .
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1. Then e Re is a left vector space over D with the dimension less than or equal to 1.
294
2. If z is a non-zero element of eRe , there exists embeddingσ : D → D satisfying
295
the property ze be =σ (e be)z for all e be ∈ D .
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3. IfdimD(e Re)=1, thenσ is an isomorphism.
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Proof (1) First we assume thate Re is non-zero withD=e ReandD =e Re. Take
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any non-zero elementer e ine Re. We show thatD(er e)= D(e Re). In fact, letese
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be an arbitrary nonzero element ofe Re. Consider the mappingφ : e R → er e R
300
defined by φ(x) = er x for all x ∈ e R. One can check that φ is a well-defined
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epimorphism. Sincee R is an indecomposable almost-injective right R-module,e R
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is uniform. Assume that Ker(φ)is nonzero. Thene R/Ker(φ)is a singular module.
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But, Im(φ)is nonsingular by the nonsingularity of R, which gives a contradiction. It
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implies Ker(φ)=0. It means thater e R ∼=e R. Similarly,ese R ∼=e R. We deduce
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that there exists an R-isomorphismσ : er e R → ese R satisfyingσ (er e) = ese.
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Call the homomorphismγ :er e R →e Rsuch thatγ (x)=σ (x)for allx ∈er e R.
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SinceRis a right CAI-ring,e Ris almoste R-injective. Then, we have the following
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two cases for the homomorphismγ.
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Case 1.σ is extended to an endmorphism ofe R:
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Take α : e R → e R an endomorphism ofe R which is an extension ofσ. Then
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ese =σ (er e)=α(er e)=eα(e)e(er e)∈ D(er e)
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Case 2.σ is not extended to an endmorphism ofe R:
313
There is a homomorphismβ :e R→ e Rsuch thatβ◦γ =ιwithι:er e R → e R
314
the inclusion. Then, we haveer e =(β◦γ )(er e)= β(ese)=eβ(e)e(ese). Since
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Dis a skew-field,ese = [eβ(e)e]−1er e ∈ D(er e).
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We deduce that D(er e) = D(e Re). Thus,e Re is a one-dimensional left vector
317
space overDife Re =0.
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(2) Letz be a non-zero element ofe Re. Then,e Re = Dz by (1). It means that
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for any e be ∈ e Re, we have ze be = uz for some u ∈ D. This defines a ring
320
monomorphismσ : D → Dsuch thatσ (e be)=u. Thus,σ (e be)z =uz= ze be
321
for alle be ∈ D.
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(3) Assume that Ris a right serial ring and dimD(e Re)= 1. Take any two non-
323
zero elementser e andese ine Re. By assumption,e Ris uniserial, we may suppose
324
ese R ≤ er e R. There ise ue ine Re such thatese =er e ue. We have thate Re
325
is a skew-field and obtainese Re =er e Re. It means thate Re is a one-dimensional
326
right vector space overD. Thene Re = Dz= z D, and soσ is an isomorphism.
327 328
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Corollary 2.17 Any I-finite right nonsingular right CAI-ring R is isomorphic to
329
Mn1(e1Re1)Mn1×n2(e1Re2) . . . Mn1×nk(e1Rek) 0 Mn2(e2Re2) . . .Mn2×nk(e2Rek) 0 0 Mn3(e3Re3) . . Mn3×nk(e3Rek)
. . . .
. . . .
0 0 . . . Mnk(ekRek)
,
330
where eiRei is a division ring, eiRei ∼=ejRej for each1≤i,j ≤ k and n1, . . . ,nk
331
are any positive integers. Furthermore, if eiRej =0, then
332
dim(eiRei(eiRej)) =1=dim((eiRej)ejRej).
333
3 On Right Noetherian Right AlmostV-rings
334
Firstly, we list some known results related to almost V-ring for the sake of complete-
335
ness.
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Theorem 3.1 [3, Theorem 3.1]The following statements are equivalent for a ring R.
337
1. R is a right almost V -ring.
338
2. For every simple R-module S, either S is injective or E(S)is projective of length
339
2.
340
Theorem 3.2 [3, Theorem 2.9]A ring R is a right almost V -ring if and only if every
341
right R-module is simple-extending.
342
Theorem 3.3 [5, Theorem 2.4]The following statements are equivalent for a ring R.
343
1. R is a right Noetherian right almost V -ring.
344
2. Every right R-module is semisimple-extending.
345
3. R = ⊕nj=1Ij, where Ij is either a Noetherian V -module with zero socle, or a
346
simple module, or an injective module of length 2.
347
4. R = I ⊕ J , where I and J are right ideals, I is Noetherian, every semisimple
348
module inσ[I]is I -injective, and every module inσ[J]is extending.
349
The following result provides a characterization of right Noetherian right almost 2
350
V-rings via almost injective semisimple modules.
351
Theorem 3.4 The following statements are equivalent for a ring R.
352
1. R is a right Noetherian right almost V -ring.
353
2. Every semisimple right R-module is almost injective and R has finite right Goldie
354
dimension.
355
3. Every semisimple right R-module is almost injective and Soc(RR)is finitely gen-
356
erated.
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Proof (1)⇒(2) By hypothesis, R has finite right Goldie dimension. Now we show
358
that every semisimple right R-module S is almost injective. Let N be any module,
359
0→ A→ N be any monomorphism for a submodule Aof N and let f : A→ Sbe
360
any non-zero homomorphism. AssumeU = E(f(A))and E(S) =U ⊕V. SinceR
361
is a right Noetherian ring,
362
U = ⊕
i∈IE(Si).
363
By Theorem3.1, either E(Si)is simple or E(Si)is projective of length 2. SinceU is
364
injective, there exists a homomorphismg : N →U such that f =g◦ι.
365
Case 1: g(N) ≤ ⊕i∈ISi. Let ω : ⊕i∈ISi → S be the natural embedding and
366
g1=ω◦g. In this case the following diagram commutes
367
0 A
f
ι N
g1
S
368
Case 2: g(N) ⊕i∈ISi. Letπi :U → E(Si)be the canonical projection. Then
369
there exists an index j ∈ I such thatπj(g(N)) Soc(E(Sj)). So thatπj(g(N)) =
370
E(Sj), since length(E(Si)≤2,for anyi ∈ I. Henceπj(g(N))is both injective and
371
projective. It follows that there exists a decomposition N = N1⊕Ker(πj ◦g), and
372
ϕ = (πj ◦g)|N1 is an isomorphism from N1 toEj. Setw1 = ϕ−1 andw2 =w1πj,
373
h1 = w2|S. Then h1 is a homomorphism from U ⊕ V to N1. Let h = h1|S and
374
π : N → N1be the canonical projection. Leta∈ A, thena =a1+a2 witha1 ∈ N1
375
anda2 ∈ K er(πj ◦g). Therefore
376
πjg(a)=πjg(a1)+πjg(a2)=πjg(a1)=πj f(a1)∈ Sj.
377
Since ϕ is isomorphic, it follows that a1 ∈ Soc(N1). Define a homomorphism ϕ :
378
Soc(N1) → Sj withθ(x) = πj f(x). Last, we putβ = πj|S andh = θ−1β. Then
379
h is a homomorphism from S to N1. Leta = x +y ∈ A, where x ∈ Soc(N1)and
380
y∈Ker(πjg). Thenπ(a)=x. Henceθ (x)=(πjf)(x), so that
381
x =θ−1(θ (x)) =θ−1(πj f(x))=θ−1(β)(f(x))=(θ−1β)(f(x))=h f(a).
382
Thereforeπ◦ι= f ◦h. In this case the following diagram commutes
383
0 A i
f
N = N1⊕ N2 π
S h N1
384
Thus,Sis an almost injective module.
385
Au th o r Pro o f
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(2)⇒(3) is clear.
386
(3)⇒(1) Assume (3). ThenRis an almost rightV-ring. LetSbe a semisimple right
387
R-module. By [4, Proposition 2.1],Sis essentially injective. Then, every semisimple
388
rightR-module is essentially injective. It follows thatR/Soc(RR)is right Noetherian,
389
by [4, Lemma 2.2]. Hence R is a right Noetherian ring since Soc(RR) is finitely
390
generated.
391
Theorem 3.5 The following statements are equivalent for a ring R.
392
1. R is an Artinian serial ring with J(R)2=0.
393
2. Every semisimple right R-module is almost injective, RR is almost injective and
394
R is a direct sum of indecomposable right ideals.
395
3. Every semisimple right R-module is almost injective, RR is almost injective and
396
Soc(RR)is finitely generated.
397
Proof First we note that ifRR is an almost injective module with finite Goldie dimen-
398
sion then Ris a direct sum of uniform right ideals. Hence, it suffices to show that (3)
399
⇒(1). Assume (3). By Theorem3.4, Ris a right Noetherian right almostV-ring, and
400
RR has a decomposition RR =e1R⊕e2R⊕ · · · ⊕enR, where eacheiRis uniform,
401
since RR is almost injective. Let e = ei, for 1 ≤ i ≤ n. We shall prove that e R is
402
a uniserial module. LetU,V be submodules ofe R. ThenU andV contain maximal
403
submodulesU1andV1, respectively, sinceRis right Noetherian. Thene R/(U1⊕V1)
404
has two distinct minimal submodules(U +V)/(U1+V)and(U +V)/(U +V1).
405
This is impossible, since e R/(U1⊕ V1)is an indecomposable module over a right
406
almostV-ring. Therefore e Ris uniserial. Assume that e R is not simple, and U is a
407
non-zero proper summodule ofe R. Then there exists a maximal submoduleU1ofU.
408
Sincee R/U1is uniform, its socle isU/U1. So length(e R/U1)= 2, since Ris a right
409
almostV-ring. HenceU is simple and length(e R) =2, and so e Ris injective. Last,
410
we get RR =e1R⊕e2R⊕ · · · ⊕enR, where eacheiRis either a simple module or
411
an injective module of length 2. By [11, 13.5,(e)⇒(g)], Ris an Artinian serial ring
412
withJ(R)2 =0.
413 414
We obtain the following result in [4, Theorem 3.1]. 3
415
Corollary 3.6 The following statements are equivalent for a ring R.
416
1. R is an Artinian serial ring with J(R)2=0.
417
2. Every right R-module is almost injective and R is a direct sum of indecomposable
418
right ideals.
419
3. Every right R-module is almost injective and Soc(RR)is finitely generated.
420
Acknowledgements Parts of this paper were written during a stay of the authors (Thuyet, Dan and Quynh)
421
in the Vietnam Institute For Advanced Study in Mathematics (VIASM) and were supported by Vietnam
422
National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-
423
2018.02. The authors would like to thank the members of VIASM for their hospitality, as well as to
424
gratefully acknowledge the received support. A. N. Abyzov was supported by the Russian Foundation for
425
Basic Research and the Government of the Republic of Tatarstan (Grant 18-41-160024). We would like to
426
thank the Reviewers for carefully reading the paper.
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