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JP Journal of Algebra, Number Theory and Applications
Volume 13, Number 2, 2009, Pages 193-208
Published Online: May 22, 2009
This paper is available online at http://www.pphmj.com
© 2009 Pushpa Publishing House
:tionClassifica jectSub sMathematic 2000 16D60, 16D99.
Keywords and phrases: closed submodules, closed weak δ-supplemented, δ-small submodule.
Received October 11, 2008
CLOSED WEAK δ-SUPPLEMENTED MODULES
Y. TALEBI and A. R. MONIRI HAMZEKOLAEI
Department of Mathematics
Faculty of Basic Science
University of Mazandaran
Babolsar, Iran
e-mail: talebi@umz.ac.ir
a.monirih@umz.ac.ir
Abstract
A module M is called closed weak δ-supplemented if for any closed
submodule N of M, there is a submodule K of M such that NKM
+
=
and .MNK δ
∩ Any direct summand of closed weak δ-supplemented
module is also closed weak δ-supplemented. Any nonsingular image of
closed weak δ-supplemented module is closed weak δ-supplemented.
Nonsingular V-rings in which all nonsingular modules are closed weak δ-
supplemented are also characterized in Section 4.
1. Introduction and Preliminaries
Throughout this article, all rings are associative and have an
identity, and all modules are unitary right R-modules.
Let M be a module. Then a submodule N of M is called an essential
submodule, denoted by ,MN e
≤
if for any nonzero submodule L of M,
.0≠NL ∩ Let K, L be submodules of M. Then L is called an essential
extension of K in M if .LK e
≤
A closed submodule N of M, denoted by
,MN c
≤ is a submodule which has no proper essential extension in M.
If NL c
≤ and ,MN c
≤ then ,ML c
≤
(see [4]).
Y. TALEBI and A. R. MONIRI HAMZEKOLAEI
194
A submodule K of M is called small in M if
M
KN
≠
+
for any
proper submodule N of M. A module M is called hollow if every proper
submodule of M is small in M.
Let M be a module. We say M is extending if any closed submodule
is direct summand. A submodule N of M is δ-small in M, denoted by
,MN δ
if MKN =+ with KM singular implies K
M
=
(see [7]).
Let N and K be submodules of M. Then N is called a δ-supplement of K in
M if MKN =+ and NKN δ
∩ and if ,MKN δ
∩ N is called a
weak δ-supplement of K in M. A module M is called δ-supplemented
if any submodule of M has a δ-supplement in M. A module M is
called weakly δ-supplemented if for any submodule N of M there is a
submodule K of M such that NK
M
+
=
and .MKN δ
∩ Clearly,
any δ-supplemented module is weakly δ-supplemented. An R-module M is
called ⊕-δ-supplemented if for every submodule N of M there is a direct
summand K of M which is a δ-supplement of N in M.
In this article, we will replace the condition of extending modules
which closed submodule is direct summand by the condition that the
closed submodule has a weak δ-supplement.
A module M is called distributive if for submodules N, L, K of M,
()
(
)
(
)
KNLNKLN ∩∩∩
+
=
+
and
()
(
)
(
)
.KNLNKLN
+
+
=
+∩∩
In Section 2, we give the definition of closed weak δ-supplemented module
and show that any direct summand of closed weak δ-supplemented
module is closed weak δ-supplemented. Let 21 MMM
+
=
be a distributive
module. Then M is closed weak δ-supplemented if and only if each i
M for
()
2,1=i is closed weak δ-supplemented.
In Section 3, we will show that any nonsingular homomorphic image
of a closed weak δ-supplemented module is closed weak δ-supplemented.
For a right nonsingular ring R, every projective right R-module is closed
weak δ-supplemented if and only if every nonsingular right R-module is
closed weak δ-supplemented.
CLOSED WEAK δ-SUPPLEMENTED MODULES 195
In Section 4, we characterize rings in which all nonsingular modules
are closed weak δ-supplemented. The relation between closed weak
δ-supplemented modules and weak δ-supplemented is given in this
section.
Let M be an R-module. Then we use
(
)
MRad to denote the Jacobson
radical of M and
()
Mδ to denote the sum of all δ-small submodules of M.
Also,
() { }
0=|∈= mrRrmr denotes the right annihilator of .Mm
∈
Now
we give some basic lemmas.
Lemma 1.1. Let M be a module and N be a submodule of M. Then
MN e
≤ if and only if for any nonzero ,Mm
∈
there is ,Rr
∈
such that
Nmr ∈ and .0≠mr
Proof. See [2, Lemma 5.19].
Lemma 1.2. Let M be an R-module and ;, MLK
≤
(1) If ML δ
and NMf →: is a homomorphism, then
(
)
;NLf δ
in particular, if ,MLk ≤
δ
then .MK δ
(2) If 11 XK δ
and ,
22 XK δ
then .
2121 XXKK
+
+
δ
(3) If ,MLK ≤≤ MK δ
and L is a direct summand of M, then
.LK δ
Proof. See [1, Lemma 1.1].
2. Closed Weak δ-supplemented Modules
Definition 2.1. A module M is called closed weak δ-supplemented if
any closed submodule of M has a weak δ-supplement in M.
It is easy to see that any extending module is closed weak
δ-supplemented and any weak δ-supplemented module is also closed
weak δ-supplemented. Clearly, any hollow module is δ-supplemented.
Thus we have the following implication:
Hollow ⇒ δ-supplemented ⇒ Weak δ-supplemented ⇒ Closed weak
δ-supplemented.
Y. TALEBI and A. R. MONIRI HAMZEKOLAEI
196
As an example of closed weak δ-supplemented module which is not
weak δ-supplemented we can consider Z as a Z-module.
Proposition 2.2. Let M be a closed weak δ-supplemented module.
Then any direct summand of M is closed weak δ-supplemented.
Proof. Let N be any direct summand of M and .NL c
≤
Since N is
closed in M, we have that .ML c
≤
Then there is a submodule K of M
such that LKM += and .MLK δ
∩ Thus
(
)
.LKNN
+
=
∩ Since
N is a direct summand of M, we have
(
)
NLKLKN δ
=
∩∩∩ by
Lemma 1.2 (3).
Lemma 2.3. Let N and L be submodules of M such that LN
+
has a
weak δ-supplement H in M and
(
)
LHN
+
∩ has a weak δ-supplement G
in N. Then GH + is a weak δ-supplement of L in M.
Proof. Since H is a weak δ-supplement of LN
+
in M and G is a
weak δ-supplement of
(
)
LHN
+
∩ in N, we have
()
(
)
,, MHLNMHLN δ
+
=
++ ∩
()
(
)
., NGLHNGLHN δ
+
=
+
+∩∩
Thus
(
)
MLGH
=
+
+
and
()
(
)
(
)
.MHLNGLHLGH δ
+
+
+
≤+ ∩∩∩
Proposition 2.4. Let 21 MMM
⊕
=
with each
(
)
2,1
=
iMi closed
weak δ-supplemented. Suppose that
(
)
icji MLMM
≤
+
∩
and
(
)
,
jcj MKLM
≤
+
∩
where K is a weak δ-supplement of
(
)
LMM ji
+
∩ in ,
i
M ,ji
≠
for any
closed submodule L of M. Then M is closed weak δ-supplemented.
CLOSED WEAK δ-SUPPLEMENTED MODULES 197
Proof. Let .ML c
≤ Then
(
)
LMMM
+
+
=
21 has trivial weak
δ-supplement 0 in M. Since
(
)
LMM
+
21 ∩ is closed in ,
1
M there is a
submodule K of 1
M such that
(
)
LMMKM
+
+
=
211 ∩
and
(
)
.
12 MLMK δ
+
∩
By Lemma 2.3, K is a weak δ-supplement of LM
+
2 in M, i.e.,
()
.
2LMKM ++= Since
(
)
,
22 MLKM c
≤
+
∩
(
)
LKM
+
∩
2 has a
weak δ-supplement J in .
2
M Again by Lemma 2.3, JK
+
is a weak δ-
supplement of L in M.
Proposition 2.5. Let 21 MMM
+
=
with 1
M closed weak δ-
supplemented and 2
M be any R-module. Suppose that for any closed
submodule N of M, .
11 MMN c
≤
∩ Then M is closed weak δ-supplemented
if and only if every closed submodule N of M with NM ⊆
/
2 has a weak
δ-supplement.
Proof.
()
⇒ This is clear.
()
⇐ Let MN c
≤ such that .
2NM ⊆ Then
NMMMM
+
=
+
=
121
has trivial weak δ-supplemented 0 in M. Since 1
MN ∩ is closed in 1
M
and 1
M is closed weak δ-supplemented, 1
MN ∩ has a weak δ-supplement
H in .
1
M By Lemma 2.3, H is a weak δ-supplement of N in M.
Let M be a nonsingular module and .MN c
≤
Then LLN c
≤
∩ for
any submodule L of M. In fact, since NM is nonsingular, so is
()
(
)
.
~NLLNNL ∩
=
+
Corollary 2.6. Let 21 MMM
+
=
be a nonsingular module with 1
M
closed weak δ-supplemented and 2
M be any R-module. Then M is closed
weak δ-supplemented if and only if every closed submodule N of M with
NM ⊆
/
2 has a weak δ-supplement.
Y. TALEBI and A. R. MONIRI HAMZEKOLAEI
198
Proposition 2.7. Let 21 MMM
⊕
=
be a distributive module. Then
M is closed weak δ-supplemented if and only if each
(
)
2,1
=
iMi is closed
weak δ-supplemented.
Proof.
()
⇒ It follows from Proposition 2.2.
()
⇐ Let .ML c
≤ Then for each i
(
)
2,1
=
i we show .
ici MML ≤∩
Suppose that .
11 MKML e
≤
≤∩ Since M is distributive, we have that
()()
(
)
.
221 LMKLMLML e∩∩∩
⊕
≤
⊕= Hence
()
(
)
(
)
,
221 LMKLMLML ∩∩∩
⊕
=
⊕
=
since .ML c
≤ So 1
MLK ∩= and 1
ML ∩ is closed in .
1
M
Thus there is a submodule i
K of i
M such that
(
)
iii MLKM ∩
+
=
and ,
ii MKL δ
∩
()
.2,1=i Hence ,
2121 LKKMMM
+
⊕
=
⊕
=
and
()()
(
)
(
)
.
212121 MMMKLKLKKL
=
⊕
⊕=⊕ δ
∩∩∩ Thus M is
closed weak δ-supplemented.
Proposition 2.8. Let M be an R-module. If any nonzero proper closed
submodule of M is maximal in M, then M is extending and hence closed
weak δ-supplemented.
Proof. See [5, Proposition 5].
Let M and L be R-modules. We call epimorphism ,: NMf → δ-small
cover if .Mfker δ
Lemma 2.9. Let NMf →: be δ-small cover and .NL δ
Then
()
.
1MLf δ
−
Proof. Suppose
()
MTLf =+
−1 with TM singular. Then
()
TfL +
N= and
()
TfN singular. Since ,NL δ
we have
(
)
(
)
TfMf
=
and this
implies .TfkerM += Thus ,TM
=
since .Mfker δ
Lemma 2.10. Let NMf →: be a δ-small cover and ML
≤
such
that
()
Lf has a weak δ-supplement in N. Then L has a weak δ-supplement
in M.
CLOSED WEAK δ-SUPPLEMENTED MODULES 199
Proof. Since
()
Lf has a weak δ-supplement in N, there is a
submodule T of N such that
(
)
NTLf
=
+
and
(
)
.NTLf δ
∩ Thus
()
MTfL =+ −1 and
() ()()
MTLffTfL δ
−− ≤∩∩ 11 by Lemma 2.9.
Therefore, L has a weak δ-supplement
()
Tf 1− in M.
Proposition 2.11. Let NMf →: be a δ-small cover and N be a
closed weak δ-supplemented module. If every nonzero closed submodule L
of M contains ker f, then M is closed weak δ-supplemented.
Proof. Let ML c
≤≠0 and suppose that
(
)
.NKLf e
≤
≤
Then
()() ()
.
11 KfLfffkerLL e−− ≤=+=
Hence
()
KfL 1−
= and
()
KLf
=
is a closed submodule of N. Thus
()
Lf
has a weak δ-supplement in N. By Lemma 2.10, L has a weak δ-
supplement in M.
Proposition 2.12. Let M be a closed weak δ-supplemented module
such that
()
MM δ is semisimple. Then ,
21 MMM
⊕
=
where 1
M is
semisimple and 2
M is a module with
(
)
.
22 MM e
≤
δ
Proof. Let 1
M be a complement of submodule
(
)
M
δ
in M. By
Lemma 5.21 in [2], we have
(
)
.
1MMM e
≤
δ
⊕
Since each complement
is closed and M is closed weak δ-supplemented, there is submodule
2
M of M such that 21 MMM
+
=
and .
21 MMM δ
∩ Since
()
,
121 MMMM δ≤ ∩∩ we have 21 MMM
⊕
=
and
(
)
(
)
2
MM
δ
=
δ
and
()()
.
212 MMMMM e∩∩ ≤δ⊕ Thus
(
)
22 MM e
≤
δ
and 21 ~MMM
=
is semisimple.
3. The Homomorphic Image
In this section, we will consider the condition for which the
homomorphic image of closed weak δ-supplemented modules is also
closed weak δ-supplemented. Any homomorphic image of weak δ-
supplemented module is weak δ-supplemented. But the homomorphic
image of extending module need not to be extending. See, Example 2.3 in
[3].
Y. TALEBI and A. R. MONIRI HAMZEKOLAEI
200
Lemma 3.1. Let NMf →: be an epimorphism of modules and
NL c
≤ and N be a nonsingular module. Then
()
.
1MLfH c
≤= −
Proof. See [5, Lemma 4].
Theorem 3.2. Let M be a closed weak δ-supplemented module.
Then any nonsingular homomorphic image of M is also closed weak
δ-supplemented.
Proof. Let NMf →: be an epimorphism of R-modules with M
closed weak δ-supplemented and N be a nonsingular module. Let
.NL c
≤ Then by Lemma 3.1,
()
.
1MLfH c
≤= − Since M is closed weak
δ-supplemented, there is a submodule K of M such that H
K
M
+
=
and
.MHK δ
∩ Hence
(
)
(
)
(
)
.LKfHfKfN
+
=
+
=
By Lemma 1.2 (1),
()()
,NLKfHKf δ
=∩∩ since .Hfker
≤
Thus N is closed weak
δ-supplemented.
Remark 3.3. In Theorem 3.2 the condition that N is nonsingular
is not necessary. For example Z is closed weak δ-supplemented as
Z-module, since Z is extending. For any prime number p, pZZZ p
=
~ is
a closed weak δ-supplemented Z-module. But p
Z is singular.
An R-module M is called singular if
(
)
,MMZ
=
where
()
=MZ
{}
,,0 Re RImIMm ≤=
|
∈ and nonsingular if
(
)
.0
=
MZ A ring R is
called right nonsingular if R
R is nonsingular and singular if R
R is
singular. Let R be a ring. Then R is right nonsingular if and only if all
right projective modules are nonsingular.
Corollary 3.4. Let M be a closed weak δ-supplemented module such
that
()
MM δ is nonsingular. Then
(
)
MM
δ
is extending.
Proof. By Theorem 3.2,
(
)
MM
δ
is closed weak δ-supplemented. Let
() ()
.MMMN cδ≤δ Then there is a submodule
(
)
MK
δ
of
()
MM δ
such that
(
)
(
)
(
)
MMMKMN
δ
=
δ
+
δ
CLOSED WEAK δ-SUPPLEMENTED MODULES 201
and
(
)
(
)
.MMMKN
δ
δ
δ
∩
It implies that
(
)
(
)
(
)
,MKMNMM
δ
⊕
δ
=
δ
since
(
)()
.0=δδ MM Thus
(
)
MN
δ
is a direct summand of
()
.MM δ
Hence
()
MM δ is extending.
Corollary 3.5. Let R be a right nonsingular ring. Then the following
are equivalent:
(1) Every nonsingular right R-module is closed weak δ-supplemented;
(2) Every projective right R-module is closed weak δ-supplemented.
A ring R is called a right closed weak δ-supplemented ring if R
R as
R-module is closed weak δ-supplemented module.
Corollary 3.6. Let R be a right nonsingular ring. Then the following
are equivalent:
(1) R is right closed weak δ-supplemented ring;
(2) Every nonsingular cyclic R-module is closed weak δ-supplemented;
(3) Every principal right ideal of R is closed weak δ-supplemented.
Lemma 3.7. Let NMf →: be an epimorphism of modules and
.NL c
≤ Then for some ,MU
≤
if
(
)
(
)
(
)
mfrmr
=
for all ,\ fkerMm
∈
then .MU c
≤
Proof. See [5, Lemma 5].
Theorem 3.8. Let NMf →: be an epimorphism of modules with M
closed weak δ-supplemented. If
(
)
(
)
(
)
mfrmr
=
for all ,\ fkerMm
∈
then
N is also closed weak δ-supplemented.
Proof. By Lemma 3.7, for any closed submodule L of N, there is a
closed submodule U of M, such that MUfker c
≤
≤
and .
~fkerUL
=
Since M is closed weak δ-supplemented,
(
)
fkerfkerK
+
is a weak
Y. TALEBI and A. R. MONIRI HAMZEKOLAEI
202
δ-supplement of fkerU in ,fkerM where K is a weak δ-supplement of
U in M. Hence N is closed weak δ-supplemented.
4. Rings in which all Nonsingular Modules are
Closed Weak δ-supplemented
In this section, we will characterize rings in which all nonsingular
modules are closed weak δ-supplemented. Also, the relations among
weak δ-supplemented, ⊕-δ-supplemented modules and closed weak
δ-supplemented modules are given.
Proposition 4.1. Let M be an R-module with
(
)
.0
=
δ
M Then the
following are equivalent:
(1) M is a closed weak δ-supplemented module;
(2) M is extending.
Proof.
()
21 ⇒ Let .MN c
≤
Then there is a submodule K of M such
that
K
NM += and .MKN δ
∩ Thus ,0
=
KN ∩ since
(
)
.0=
δ
M
It implies N is a direct summand of M. Hence M is extending.
()
12 ⇒ This is clear.
Corollary 4.2. Let R be a ring with
(
)
.0
=
δ
R
R Then the following are
equivalent:
(1) R is a closed weak δ-supplemented ring;
(2) R is an extending ring.
A ring R is called a right V-ring if every simple right R-module is
injective. Equivalently, a ring R is a right V-ring if and only if
()
0=MRad for all right R-modules M.
Theorem 4.3. Let R be a right nonsingular ring with
(
)
0
=
δ
M for all
right R-modules M. Then the following are equivalent:
(1) Every nonsingular right R-module M is closed weak δ-supplemented;
(2) Every projective right R-module M is closed weak δ-supplemented;
CLOSED WEAK δ-SUPPLEMENTED MODULES 203
(3) Every nonsingular right R-module M is extending;
(4) Every nonsingular right R-module M is projective.
Proof. (1) ⇔ (2) By Corollary 3.5.
(1) ⇔ (3) It follows from Proposition 4.1.
(2) ⇒ (4) Let M be a nonsingular module. Then there is a projective
module P, such that NPM =
~ for some submodule N of P. Since P is
nonsingular, we have that N is a closed submodule of P. By (2), P is
closed weak δ-supplemented, hence P is extending by Proposition 4.1.
Thus N is a direct summand of P and therefore M is projective.
(4) ⇒ (1) Let M be a nonsingular module and N be a closed
submodule of M. Then NM is nonsingular, hence by (4), is projective.
Thus N is a direct summand of M. Hence M is extending and closed weak
δ-supplemented.
Combining with Theorem 5.23 in [4] and the Theorem 4.3, we have
Corollary 4.4. The following are equivalent for a ring R with
()
0=δ M for all R-modules M:
(1) R is right nonsingular and every nonsingular right module M is
closed weak δ-supplemented;
(2) R is right nonsingular and all nonsingular right R-modules are
projective;
(3) R is right nonsingular and every projective right R-module is
closed weak δ-supplemented;
(4) R is right nonsingular and every nonsingular right R-module M is
extending;
(5) R is left nonsingular and every nonsingular left R-module M is
closed weak δ-supplemented;
(6) R is left nonsingular and all nonsingular left R-modules are
projective;
(7) R is left nonsingular and every projective left R-module is closed
weak δ-supplemented;
Y. TALEBI and A. R. MONIRI HAMZEKOLAEI
204
(8) R is left nonsingular and every nonsingular left R-module M is
extending;
(9) R is right and left hereditary, right and left artinian, and the
maximal right and left quotient rings of R coincide.
We recall a module M, uniserial if its lattice of submodules is linearly
ordered by inclusion.
Since any regular ring is left and right nonsingular, combining with
Exercises 5.C.17, 21 in [4], we have
Corollary 4.5. Let R be a commutative regular ring such that every
nonsingular module is closed weak δ-supplemented. Then R is semisimple
and every nonsingular R-module is a direct sum of uniserial modules.
Next, we will study the relation between closed weak δ-supplemented
modules and weak δ-supplemented modules.
Lemma 4.6. Let NMf →: and KNg →: be δ-small covers. Then
KMgof →: is also a δ-small cover.
Proof. Let
()
MTgofker
=
+ with TM singular for some .MT ≤
Then
()
,NTfgker =+ since
(
)
(
)
.gkergofkerf ⊆ Also,
(
)
TfN is singular,
thus
() ()
.TfMfN == It implies that .fkerTM
+
=
Therefore ,TM =
since .Mfker δ
Lemma 4.7. Let M be closed weak δ-supplemented and N be a closed
submodule of M. Suppose that .MT δ
Then there is a submodule K of
M such that
T
N
K
N
K
M
+
+
=+= and ,MNK δ
∩
(
)
.MTNK δ
+
∩
Proof. Since M is closed weak δ-supplemented and N is a closed
submodule of M, then there is a submodule K of M such that N
K
M
+
=
and .MNK δ
∩ Let
(
)
(
)
,: KMNMMf
⊕
→ which is defined by
() ( )
,, KmNmmf ++= and
(
)
(
)
(
)
(
)()
,: KMTNMKMNMg
⊕
+
→
⊕
which is defined by
()
(
)
.,, KmTNmKmNmg
+
′
+
+
=
+
′
+ Since =M
,KN + then we have that f is an epimorphism and that =fker
.MKN δ
∩ Since
(
)
(
)
0
⊕
+
=
NTNgker and
(
)()
TNTN π=
+
CLOSED WEAK δ-SUPPLEMENTED MODULES 205
,NM
δ
where NMM →π : is the canonical epimorphism, we have
that g is a δ-small cover. So by Lemma 4.6, gof is a δ-small cover. Thus
()( )
.MKNTgofker δ
+= ∩ Clearly,
(
)
.KTNM
+
+
=
Proposition 4.8. Let M be a singular R-module. Suppose that for any
submodule N of M, there is a closed submodule L (depending on N) of M
such that either
T
LN += or
T
NL
′
+
=
for some ., MTT δ
′
Then
M is weak δ-supplemented if and only if M is closed weak δ-supplemented.
Proof. Suppose that M is closed weak δ-supplemented and N is any
submodule of M.
Case 1. Suppose that there is a closed submodule L such that =N
T
L+ for some .MT δ
Then this is a consequence of the Lemma 4.7.
Case 2. Suppose that there is a closed submodule L of M such that
T
NL ′
+= for some .MT δ
′ Since M is closed weak δ-supplemented,
there is a submodule K of M such that L
K
M
+
=
and .MLK δ
∩ So
,TNKM ′
++= hence ,NKM
+
=
since M is singular and .MT δ
′
Also, .MLKNK δ
≤∩∩ Thus M is weak δ-supplemented.
The converse is trivial.
Definition 4.9. Let M be an R-module. Then M is called refinable if
for submodules U, V with ,MVU
=
+
there is a direct summand U′ of
M with UU ⊆
′ and ,MVU
=
+
′ (see [6]).
Theorem 4.10. Let M be a refinable singular module. Suppose that
for any submodule N of M, there is a closed submodule L (depending on
N) of M such that either
T
LN
+
=
or
T
NL
′
+
=
for some ., MTT δ
′
Then the following are equivalent:
(1) M is ⊕-δ-supplemented;
(2) M is δ-supplemented;
(3) M is weak δ-supplemented;
(4) M is closed weak δ-supplemented.
Y. TALEBI and A. R. MONIRI HAMZEKOLAEI
206
Proof. It is clear that (1) ⇒ (2) ⇒ (3) ⇒ (4).
(4) ⇒ (1) Let N be any submodule of M.
Case 1. Suppose that there is a closed submodule L of M such that
T
LN += for some .MT δ
Since M is closed weak δ-supplemented,
there is a submodule K of M such that
K
L
M
+
=
and .MLK δ
∩
Hence
K
N
K
LM +=+= and MKN δ
∩ by Lemma 4.7. Since M
is refinable, there is a direct summand U of M such that KU ⊆ and
.NUM += So .MNKNU δ
⊆∩∩ As U is a direct summand of
M, we have that ,UNU δ
∩ which shows that M is ⊕-δ-supplemented.
Case 2. Suppose that there is a closed submodule L of M such that
T
NL ′
+= for some .MT δ
′ Since M is closed weak δ-supplemented,
there is a submodule K of M such that
K
L
M
+
=
and .MKL δ
∩
Thus ,KNKTNKLM
+
=
+
′
+=+= since M is singular and
.MKN δ
∩ As M is refinable, then there is a direct summand U of M
such that KU ⊆ and .NUM
+
=
Therefore
.MNKNU δ
⊆∩∩
Since U is a direct summand of M, ,UNU δ
∩ and hence M is ⊕-δ-
supplemented.
Corollary 4.11. Let R be a ring and M be a refinable singular module
with
()
.0=δ M Suppose that for any submodule N of M, there is a closed
submodule L (depending on N) of M such that either
T
LN +
=
or
T
NL ′
+= for some ., MTT δ
′
Then the following are equivalent:
(1) M is ⊕-δ-supplemented;
(2) M is δ-supplemented;
(3) M is weak δ-supplemented;
(4) M is closed weak δ-supplemented;
(5) M is extending.
CLOSED WEAK δ-SUPPLEMENTED MODULES 207
Lemma 4.12. Let U and K be submodules of M such that K is singular
and a weak δ-supplement of a maximal submodule N of M. If UK + has
a weak δ-supplement X in M, then U has a weak δ-supplement in M.
Proof. Since X is a weak δ-supplement of UK
+
in M,
XUKM
+
+
=
and
(
)
.MUKX δ
+
∩
If
()
,MNKUXK δ
⊆+ ∩∩ then
()
(
)
(
)
,MUXKUKXXKU δ
+
+
+
⊆+ ∩∩∩
hence X
K
+ is a weak δ-supplement of U in M.
Suppose that
()
.NKUXK ∩∩ ⊆
/
+
Since
(
)
(
)
NNKNKK +
=
~
∩
,NM= NK ∩ is a maximal submodule of K. Therefore,
(
)
KUXKNK
=
+
+
∩∩
and since ,MNK δ
∩ we have
(
)
.NKUXUXKNKUXKUXM ∩∩∩
+
+
=
+
+
+
+=++=
Since
()
(
)()
UXKKUXM
+
=+ ∩
~ and K is singular, we have
UXM
+
=
and since
()
,MXUKXU δ
+≤ ∩∩ X is a weak δ-supplement of U in
M.
As a result of Lemma 4.12, we have
Theorem 4.13. Suppose that for any submodule U of M, there is a
singular submodule K of M, which is a weak δ-supplement of some
maximal submodule N of M, such that UK
+
is closed in M. Then M is
closed weak δ-supplemented if and only if M is weak δ-supplemented.
Y. TALEBI and A. R. MONIRI HAMZEKOLAEI
208
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