A note on noncosingular lifting modules
Let R be a right perfect ring. Let M be a noncosingular lifting module that does not have any relatively projective component. Then M has finite hollow dimension. Нехай R — праве досконале кiльце, а M — некосингулярний модуль iз властивiстю пiдняття, що не має жодної вiдносно проективної компоненти....
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| Опубліковано в: : | Український математичний журнал |
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| Дата: | 2012 |
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Український математичний журнал
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| Цитувати: | A note on noncosingular lifting modules / T. Amouzegar Kalati, D. Keskin Tütüncü // Український математичний журнал. — 2012. — Т. 64, № 11. — С. 1572-1574. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859613459085262848 |
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| author | Amouzegar Kalati, T. Keskin Tütüncü, D. |
| author_facet | Amouzegar Kalati, T. Keskin Tütüncü, D. |
| citation_txt | A note on noncosingular lifting modules / T. Amouzegar Kalati, D. Keskin Tütüncü // Український математичний журнал. — 2012. — Т. 64, № 11. — С. 1572-1574. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Український математичний журнал |
| description | Let R be a right perfect ring. Let M be a noncosingular lifting module that does not have any relatively projective component. Then M has finite hollow dimension.
Нехай R — праве досконале кiльце, а M — некосингулярний модуль iз властивiстю пiдняття, що не має жодної вiдносно проективної компоненти. Тодi M має скiнченну дуальну розмiрнiсть Голдi.
|
| first_indexed | 2025-11-28T16:13:07Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
T. Amouzegar Kalati (Quchan Inst. Engineering and Technology, Iran),
D. Keskin Tütüncü (Hacettepe Univ., Ankara, Turkey)
A NOTE ON NONCOSINGULAR LIFTING MODULES
ПРО НЕКОСИНГУЛЯРНI МОДУЛI IЗ ВЛАСТИВIСТЮ ПIДНЯТТЯ
Let R be a right perfect ring. Let M be a noncosingular lifting module which does not have any relatively projective
component. Then M has finite hollow dimension.
Нехай R — праве досконале кiльце, а M — некосингулярний модуль iз властивiстю пiдняття, що не має жодної
вiдносно проективної компоненти. Тодi M має скiнченну дуальну розмiрнiсть Голдi.
1. Introduction. Throughout this paper all rings are associative with identity and modules are unitary
right modules. A module M is said to have finite hollow dimension if there exists an epimorphism
from M to a finite direct sum of n hollow factor modules with small kernel. A module M is
called lifting if for every A ≤ M, there exists a direct summand B of M such that B ⊆ A and
A/B �M/B. A module M is amply supplemented and every coclosed submodule of M is a direct
summand of M if and only if M is lifting by [1] (22.3(d)). In [5], Talebi and Vanaja defined Z(M)
as follows:
Z(M) = Re (M,S) =
⋂{
Ker(g) | g ∈ Hom (M,L), L ∈ S
}
,
where S denotes the class of all small modules.
They called M a cosingular (noncosingular) module if Z(M) = 0 (Z(M) =M ).
In this note, as we state in the abstract, we prove the following main theorem:
Let R be a right perfect ring. Let M be a noncosingular lifting module which does not have any
relatively projective component. Then M has finite hollow dimension.
For all undefined notions we refer to [1].
2. Results. An R-module M is called dual Rickart if, for any element φ ∈ S = End(M),
Imφ = eM, where e2 = e ∈ S.
Lemma 2.1. Let M = ⊕i∈NMi be a dual Rickart module and let (fi : Mi →Mi+1)N be a se-
quence of homomorphisms. Then for any finitely many elements a1, a2, . . . , an ∈M1, there exist some
r ∈ N and a homomorphism h : Mr+1 → Mr such that fr−1fr−2 . . . f1(ak) = hfrfr−1 . . . f1(ak)
for k = 1, 2, . . . , n.
In particular, if M1 is finitely generated, then fr−1fr−2 . . . f1 = hfrfr−1 . . . f1.
Proof. It is easy to see by [6] (43.3(3)).
In [3], Keskin Tütüncü and Tribak introduced the concept of dual Baer modules. A module M is
called a dual Baer module if for every right ideal I of S,
∑
φ∈I
Imφ is a direct summand of M. It
is clear that every dual Baer module is dual Rickart.
Lemma 2.2. Let M = ⊕∞i=1Mi, where each Mi is local noncosingular. If, for each i, there is
an epimorphism fi : Mi −→Mi+1, which is not an isomorphism, then M is not lifting.
c© T. AMOUZEGAR KALATI, D. KESKİN TÜTÜNCÜ, 2012
1572 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 11
A NOTE ON NONCOSINGULAR LIFTING MODULES 1573
Proof. Let M = ⊕∞i=1Mi be a lifting module and (fi : Mi →Mi+1)N be a sequence of epimor-
phisms, which are non-isomorphisms. By [3] (Theorem 2.14) and Lemma 2.1, there exist an r ∈ N
and a homomorphism h : Mr+1 → Mr such that fr−1fr−2 . . . f1 = hfrfr−1 . . . f1. Since all fi are
epimorphisms, we have hfr = 1Mr . Hence fr is an isomorphism, a contradiction.
Lemma 2.2 is proved.
Recall that a family of modules {Mi | i ∈ I} is called locally semi-T-nilpotent if, for any
countable set of non-isomorphisms {fn : Min → Min+1} with all in distinct in I, and for any
x ∈Mi1 , there exists k (depending on x) such that fk . . . f1(x) = 0 (see [4]).
Corollary 2.1. Let M be a noncosingular lifting module such that M = ⊕∞i=1Mi, where each
Mi is local for all i ∈ N. Then the family {Mi | i ∈ N} is locally semi-T-nilpotent.
Proof. Consider any infinite sequence of non-isomorphisms fn
Mi1
f1→Mi2
f2→ . . . Min
fn→ . . . .
It is obvious that fn is an epimorphism for all n ≥ 1. By Lemma 2.2, it is easy to see that the family
{Mi | i ∈ N} is locally semi-T -nilpotent.
Lemma 2.3. Let U and V be noncosingular hollow modules such that the module U ⊕ V is
lifting. Then there exists an epimorphism from U to V or V is U -projective.
Proof. Let M = U ⊕ V, M1 = U ⊕ 0 and M2 = 0 ⊕ V. Hence M = M1 ⊕M2. Suppose that
there does not exist any epimorphism from U to V, i.e., from M1 to M2. We will show that V is U -
projective. Let N be any nonzero proper submodule of M such that M = N+M1. Since M is lifting,
there exists a direct summand K of M such that K ≤ N and N/K � M/K. Let M = K ⊕ K ′
for some submodule K ′ of M. Note that K and K ′ are hollow. Since M = K +M1, we have an
epimorphism from M/K ′ to M2. If K ′ + M1 = M, then we have an epimorphism from M1 to
M/K ′. So we have an epimorphism from M1 to M2, a contradiction. Thus K ′ +M1 6= M. Hence
(K ′ +M1)/K
′ �M/K ′. Since every small module is cosingular, (K ′ +M1)/K
′ is cosingular. On
the other hand, (K ′ +M1)/K
′ ∼= M1/(K
′ ∩M1) is noncosingular. Hence K ′ = K ′ +M1 and so
M1 ≤ K ′. Thus M = K ⊕M1. By [6] (41.14), M2 is M1-projective, i.e., V is U -projective.
Theorem 2.1. Let R be a right perfect ring. Let M be a noncosingular lifting module which
does note have any relatively projective component. Then M has finite hollow dimension.
Proof. By [3] (Theorem 2.14 and Corollary 2.6(ii)), there exists an index set I and hollow
submodules Mi, i ∈ I, such that M = ⊕i∈IMi. Suppose that I is infinite. For all distinct i, j in I,
Mi ⊕Mj is lifting and hence by Lemma 2.3, there exists an epimorphism from Mi to Mj or Mj
is Mi-projective. By hypothesis, there exists an epimorphism from Mi to Mj . Now by Lemma 2.2,
there exists an infinite subset J of I such that Mi
∼=Mj for all i, j ∈ J since ⊕i∈IMi is lifting.
Let i ∈ J. Suppose that φ : Mi −→ Mi is a nonzero homomorphism. Since Mi is noncosingular
and hollow, φ is an epimorphism. Suppose φ is not an isomorphism. Then for each i, j ∈ J, φ induces
an epimorphism φij : Mi −→ Mj which is not an isomorphism, contradicting Lemma 2.2. Thus φ
is an isomorphism. It follows that the ring End(Mi) of endomorphisms of Mi is a division ring,
and by [2] (Lemma 1), Mi is Mi
∼= Mj-projective, a contradiction. Therefore, M has finite hollow
dimension.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 11
1574 T. AMOUZEGAR KALATI, D. KESKİN TÜTÜNCÜ
Corollary 2.2. Let R be a right perfect ring. Let M be a noncosingular lifting module which
does not have any relatively projective component. Then M satisfies ACC equivalently, DCC on
supplements.
Proof. By Theorem 2.1 and [1] (20.34).
Finally, we give the following:
Proposition 2.1. Let R be a right perfect ring and let M =
∏∞
i=1
Mi, where each Mi is
hollow noncosingular. If, for each i, there is an epimorphism fi : Mi+1 −→ Mi, which is not an
isomorphism, then M is not lifting.
Proof. Assume that g1 : P1 −→ M1 is a projective cover of M1. Since P1 is projective, there
exists a homomorphism g2 : P1 −→M2 such that f1g2 = g1. Clearly, g2 is epic. Then for each i, we
may define inductively, gi : P1 −→ Mi so that figi+1 = gi and all gi are epic. Note that P1 and all
Mi are local and so cyclic. Now we have the strictly descending sequence since each fi is not monic
for each i:
P1 ⊃ Ker g1 ⊃ Ker g2 ⊃ . . . .
Define the homomorphism χ : P1 −→ M by χ(y) = (gi(y))i∈I (y ∈ P1). Let Imχ = K. Then
K is local and nonzero. Assume that K = xR for some nonzero element x ∈ K. We can suppose
without loss of generality that x = (0, 0, . . . , 0, xn+2, xn+3, . . .) for some positive integer n. Then
x ∈ N =
∏∞
n+2
Mi. So K ⊆ N. Note that K is coclosed in M by [5] (Lemma 2.3(2)).
Now, let M = K ⊕ K ′ for some submodule K ′ of M and let y ∈ Ker gn. Consider t =
= (0, 0, . . . , 0, gn+1(y), gn+2(y), . . .) ∈ M =
∏∞
i=1
Mi. Then t = t1 + t2 for some t1 ∈ K and
t2 ∈ K ′. Then t2 = t − t1 ∈ K ∩ K ′ = 0. So t = t1 ∈ K ⊆ N. Thus gn+1(y) = 0 and so
y ∈ Ker gn+1. It follows that Ker gn = Ker gn+1, a contradiction. Therefore K is not a direct
summand of M and M is not a lifting module.
Acknowledgments. This work has been done during a visit of the first author to the second
author in the Department of Mathematics, Hacettepe University in 2011, and she wishes to thank the
department for their kind hospitality. The first author also wishes to thank the Ministry of Science of
Iran for the support.
1. Clark J., Lomp C., Vanaja N., Wisbauer R. Lifting modules // Frontiers in Math. – Birkhäuser Verlag, 2006.
2. Keskin D. Finite direct sums of (D1)-modules // Tr. J. Math. – 1998. – 22, № 1. – P. 85 – 91.
3. Keskin Tütüncü D., Tribak R. On dual Baer modules // Glasgow Math. J. – 2010. – 52. – P. 261 – 269.
4. Mohamed S. H., Müller B. J. Continuous and discrete modules // London Math. Soc. Lect. Notes. Ser. 147. –
Cambridge: Cambridge Univ. Press, 1990.
5. Talebi Y., Vanaja N. The torsion theory cogenerated by M -small modules // Communs Algebra. – 2002. – 30, № 3.
– P. 1449 – 1460.
6. Wisbauer R. Foundations of module and ring theory. – Reading: Gordon and Breach, 1991.
Received 26.12.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 11
|
| id | nasplib_isofts_kiev_ua-123456789-165409 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-3190 |
| language | English |
| last_indexed | 2025-11-28T16:13:07Z |
| publishDate | 2012 |
| publisher | Український математичний журнал |
| record_format | dspace |
| spelling | Amouzegar Kalati, T. Keskin Tütüncü, D. 2020-02-13T12:23:40Z 2020-02-13T12:23:40Z 2012 A note on noncosingular lifting modules / T. Amouzegar Kalati, D. Keskin Tütüncü // Український математичний журнал. — 2012. — Т. 64, № 11. — С. 1572-1574. — Бібліогр.: 6 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165409 512.5 Let R be a right perfect ring. Let M be a noncosingular lifting module that does not have any relatively projective component. Then M has finite hollow dimension. Нехай R — праве досконале кiльце, а M — некосингулярний модуль iз властивiстю пiдняття, що не має жодної вiдносно проективної компоненти. Тодi M має скiнченну дуальну розмiрнiсть Голдi. en Український математичний журнал Український математичний журнал Короткі повідомлення A note on noncosingular lifting modules Про некосингулярнi модулi iз властивiстю пiдняття Article published earlier |
| spellingShingle | A note on noncosingular lifting modules Amouzegar Kalati, T. Keskin Tütüncü, D. Короткі повідомлення |
| title | A note on noncosingular lifting modules |
| title_alt | Про некосингулярнi модулi iз властивiстю пiдняття |
| title_full | A note on noncosingular lifting modules |
| title_fullStr | A note on noncosingular lifting modules |
| title_full_unstemmed | A note on noncosingular lifting modules |
| title_short | A note on noncosingular lifting modules |
| title_sort | note on noncosingular lifting modules |
| topic | Короткі повідомлення |
| topic_facet | Короткі повідомлення |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/165409 |
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