H -supplemented modules with respect to a preradical
Let M be a right R-module and τ a preradical. We call M τ-H-supplemented if for every submodule A of M there exists a direct summand D of M such that (A+D)/D⊆τ(M/D) and (A+D)/A⊆τ(M/A). Let τ be a cohereditary preradical. Firstly, for a duo module M=M₁⊕M₂ we prove that M is τ-H-supplemented if and on...
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| Zitieren: | H -supplemented modules with respect to a preradical/ Yahya Talebi, A. R. Moniri Hamzekolaei, Derya Keskin Tutuncu // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 116–131. — Бібліогр.: 16 назв. — англ. |
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irk-123456789-1548212019-06-17T01:31:04Z H -supplemented modules with respect to a preradical Yahya Talebi A. R. Moniri Hamzekolaei Derya Keskin Tutuncu Let M be a right R-module and τ a preradical. We call M τ-H-supplemented if for every submodule A of M there exists a direct summand D of M such that (A+D)/D⊆τ(M/D) and (A+D)/A⊆τ(M/A). Let τ be a cohereditary preradical. Firstly, for a duo module M=M₁⊕M₂ we prove that M is τ-H-supplemented if and only if M₁ and M₂ are τ-H-supplemented. Secondly, let M=⊕ⁿi=1Mi be a τ-supplemented module. Assume that Mi is τ-Mj-projective for all j>i. If each Mi is τ-H-supplemented, then M is τ-H-supplemented. We also investigate the relations between τ-H-supplemented modules and τ-(⊕-)supplemented modules. 2011 Article H -supplemented modules with respect to a preradical/ Yahya Talebi, A. R. Moniri Hamzekolaei, Derya Keskin Tutuncu // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 116–131. — Бібліогр.: 16 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16S90, 16D10, 16D70, 16D99. http://dspace.nbuv.gov.ua/handle/123456789/154821 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let M be a right R-module and τ a preradical. We call M τ-H-supplemented if for every submodule A of M there exists a direct summand D of M such that (A+D)/D⊆τ(M/D) and (A+D)/A⊆τ(M/A). Let τ be a cohereditary preradical. Firstly, for a duo module M=M₁⊕M₂ we prove that M is τ-H-supplemented if and only if M₁ and M₂ are τ-H-supplemented. Secondly, let M=⊕ⁿi=1Mi be a τ-supplemented module. Assume that Mi is τ-Mj-projective for all j>i. If each Mi is τ-H-supplemented, then M is τ-H-supplemented. We also investigate the relations between τ-H-supplemented modules and τ-(⊕-)supplemented modules. |
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Yahya Talebi A. R. Moniri Hamzekolaei Derya Keskin Tutuncu |
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Yahya Talebi A. R. Moniri Hamzekolaei Derya Keskin Tutuncu H -supplemented modules with respect to a preradical Algebra and Discrete Mathematics |
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Yahya Talebi A. R. Moniri Hamzekolaei Derya Keskin Tutuncu |
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Yahya Talebi |
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H -supplemented modules with respect to a preradical |
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H -supplemented modules with respect to a preradical |
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H -supplemented modules with respect to a preradical |
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H -supplemented modules with respect to a preradical |
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H -supplemented modules with respect to a preradical |
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h -supplemented modules with respect to a preradical |
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Інститут прикладної математики і механіки НАН України |
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H -supplemented modules with respect to a preradical/ Yahya Talebi, A. R. Moniri Hamzekolaei, Derya Keskin Tutuncu // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 116–131. — Бібліогр.: 16 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT yahyatalebi hsupplementedmoduleswithrespecttoapreradical AT armonirihamzekolaei hsupplementedmoduleswithrespecttoapreradical AT deryakeskintutuncu hsupplementedmoduleswithrespecttoapreradical |
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2025-07-14T06:54:21Z |
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2025-07-14T06:54:21Z |
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1837604339126894592 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 12 (2011). Number 1. pp. 116 – 131
c© Journal “Algebra and Discrete Mathematics”
H -supplemented modules
with respect to a preradical
Yahya Talebi, A. R. Moniri Hamzekolaei
and Derya Keskin Tütüncü
Communicated by R. Wisbauer
Abstract. Let M be a right R-module and τ a preradical.
We callM τ -H-supplemented if for every submodule A ofM there ex-
ists a direct summand D of M such that (A+D)/D ⊆ τ(M/D) and
(A+D)/A ⊆ τ(M/A). Let τ be a cohereditary preradical. Firstly, for
a duo module M = M1⊕M2 we prove that M is τ -H-supplemented
if and only if M1 and M2 are τ -H-supplemented. Secondly, let
M = ⊕n
i=1
Mi be a τ -supplemented module. Assume that Mi is
τ -Mj-projective for all j > i. If each Mi is τ -H-supplemented, then
M is τ -H-supplemented. We also investigate the relations between
τ -H-supplemented modules and τ -(⊕-)supplemented modules.
Introduction
Throughout this paper, R denotes an associative ring with identity and
modules are unital right R-modules. We use N ≤ M and N ≤d M to
signify that N is a submodule and a direct summand of M , respectively.
A functor τ from the category of the right R-modules Mod − R to
itself is called a preradical if it satisfies the following properties:
i) For any R-module M , τ(M) is a submodule of an R-module M,
ii) If f : M ′ → M is an R-module homomorphism, then f(τ(M ′)) ⊆
τ(M) and τ(f) is the restriction of f to τ(M ′).
The authors would like to thank Prof. R. Wisbauer and the referee for their helpful
comments and carefully reading this article.
2000 Mathematics Subject Classification: 16S90, 16D10, 16D70, 16D99.
Key words and phrases: H-supplemented module, τ -H-supplemented module,
τ -lifting module.
Y. Talebi, A. R. Moniri Hamzekolaei, D. Keskin 117
It is well known if K is a direct summand of M, then τ(K) = τ(M)∩K
for a preradical τ . A preradical τ is said to be cohereditary if, for every
M ∈ Mod−R and every submodule N of M , τ(M/N) = (τ(M) +N)/N .
We refer to [3] for details concerning radicals and preradicals. In this
paper, τ will be a preradical unless otherwise stated. Recall that a module
M has the Summand Sum Property, (SSP ) if the sum of any two direct
summands of M is again a direct summand (see [4]).
Let M be a module. A submodule X of M is called fully invariant,
if for every f ∈ End(M), f(X) ⊆ X. The module M is called a duo
module, if every submodule of M is fully invariant. The submodule A of
M is called projection invariant in M if f(A) ⊆ A, for any idempotent
f ∈ End(M). A submodule K of M is called small in M (denoted by
K ≪ M) if N +K 6= M for any proper submodule N of M .
Lifting modules were defined and studied by many authors. H-supp-
lemented modules were introduced in [11] as a generalization of lifting
modules. According to [11], a module M is called H-supplemented if for
every submodule A of M there exists a direct summand D of M such
that A + X = M if and only if D + X = M for any submodule X of
M . For more information about H-supplemented modules we refer the
reader to [8], [10] and [11]. A module M is called ⊕-supplemented if for
every submodule N of M there exists a direct summand D of M such
that M = N + D and N ∩ D ≪ D. According to [15], a module M is
semiperfect if every factor module of M has a projective cover. By [15,
41.14 and 42.1], if P is projective, then P is semiperfect if and only if
for every submodule K of P there exists a decomposition K = A ⊕ B
such that A is a direct summand of P and B ≪ P . By [5, Lemma 1.2] a
projective module is ⊕-supplemented if and only if it is semiperfect.
In [2], for a radical τ , Al-Takhman, Lomp and Wisbauer defined and
studied the concept of τ -lifting, τ -supplemented and τ -semiperfect modules.
Following [2], a module M is called τ -lifting if every submodule N of M
has a decomposition N = A⊕B such that A is a direct summand of M
and B ⊆ τ(M) and they call M τ -supplemented if for every submodule
N of M there exists a submodule K of M such that N +K = M and
N ∩K ⊆ τ(K) (In this case K is called a τ -supplement of N in M). They
call a module M τ -semiperfect if for every submodule N of M , M/N has
a projective τ -cover. In this paper we define τ -H-supplemented modules
and investigate the general properties of such modules.
In Section 1 we will define τ -H-supplemented modules and give an
equivalent condition for such modules. Also we obtain some conditions
which under the factor module of a τ -H-supplemented module will be τ -H-
supplemented. Let M be a τ -H-supplemented module for a cohereditary
preradical τ . Then
118 H -supplemented modules
(1) If M is a distributive module, then M/X is τ -H-supplemented for
every submodule X of M .
(2) Let N ≤ M such that for each decomposition M = M1 ⊕M2 we have
N = (N ∩M1)⊕ (N ∩M2). Then M/N is τ -H-supplemented.
(3) Let X be a projection invariant submodule of M . Then M/X is τ -H-
supplemented. In particular, for every fully invariant submodule A of M ,
M/A is τ -H-supplemented (Corollary 1).
In Section 2 we will study direct summands of τ -H-supplemented
modules. We show that, if τ is a cohereditary preradical, every direct
summand of a τ -H-supplemented module with SSP is τ -H-supplemented
(Theorem 2).
In Section 3 we will study direct sums of τ -H-supplemented mod-
ules. Let τ be a cohereditary preradical. Let M = M1 ⊕ M2 be a duo
module. Then M is τ -H-supplemented if and only if M1 and M2 are
τ -H-supplemented (Theorem 4). Let τ be a cohereditary preradical. Let
M = ⊕n
i=1Mi be a τ -supplemented module. Assume that Mi is τ -Mj-
projective for all j > i. If each Mi is τ -H-supplemented, then M is
τ -H-supplemented (Corollary 4).
In Section 4 we will obtain the relations between τ -H-supplemented
modules and the other modules. Let τ be a cohereditary preradical. Let
M be a projective module such that every τ -supplement submodule of M
is a direct summand. The following are equivalent: (Theorem 6)
(1) M is τ -supplemented;
(2) M is τ -lifting;
(3) M is amply τ -supplemented;
(4) M is τ -H-supplemented and τ(M) is QSL in M ;
(5) M is τ -⊕-supplemented.
1. Factor modules of τ-H-supplemented modules
In this section we will define τ -H-supplemented modules and give an
equivalent condition for a module to be τ -H-supplemented. Also we
investigate some conditions for factor modules of a τ -H-supplemented
module to be τ -H-supplemented.
Keskin Tütüncü, Nematollahi and Talebi give equivalent conditions
for a module to be H-supplemented (see [8, Theorem 2.1]). Now we give
the definition of a τ -H-supplemented module based on their definition.
Definition 1. Let M be a module. Then M is τ -H-supplemented in
case for every A ≤ M there exists a direct summand D of M such that
(A+D)/A ⊆ τ(M/A) and (A+D)/D ⊆ τ(M/D).
Y. Talebi, A. R. Moniri Hamzekolaei, D. Keskin 119
In this paper, τ -H-supplement will mean that a direct summand D of
M exists with the stated inclusions in Definition 1. The definition shows
that every τ -lifting module is τ -H-supplemented.
Next we give an equivalent condition for a module to be τ -H-supplem-
ented.
Proposition 1. Let M be a module. Then M is τ -H-supplemented if
and only if for each A ≤ M there exists a direct summand D of M and
a submodule X of M such that A ⊆ X, D ⊆ X, X/A ⊆ τ(M/A) and
X/D ⊆ τ(M/D).
Proof. ( ⇒ ) It is clear.
( ⇐ ) Let A ≤ M . By assumption, there exist a direct summand D of M
and X ≤ M such that (A+D)/A ⊆ X/A ⊆ τ(M/A) and (A+D)/D ⊆
X/D ⊆ τ(M/D). Hence M is τ -H-supplemented.
A factor module of a τ -H-supplemented module need not be τ -H-
supplemented in general. Before giving a counter example to the fact
that a factor module of a τ -H-supplemented module need not be τ -
H-supplemented in case τ = Rad, we have to mention the following
definitions:
A commutative ring R is a valuation ring if it satisfies one of the
following three equivalent conditions:
(1) for any two elements a and b, either a divides b or b divides a.
(2) the ideals of R are linearly ordered by inclusion.
(3) R is a local ring and every finitely generated ideal is principal.
A module M is called finitely presented if M ∼= F/K for some finitely
generated free module F and finitely generated submodule K of M .
Example 1. Let R be a commutative local ring which is not a valuation
ring and let n ≥ 2. By [16, Theorem 2], there exists a finitely presented
indecomposable module M = R(n)/K which cannot be generated by fewer
than n elements. By [5, Corollary 1.6], R(n) is ⊕-supplemented and hence
H-supplemented by [9, Proposition 2.1]. Being finitely generated, R(n) is
Rad-H-supplemented. Since M is not cyclic, it is not ⊕-supplemented, and
hence not H-supplemented. Since M is finitely generated, it is not Rad-
H-supplemented. (Note that since R/JacR is semisimple, the preradical
Rad is also cohereditary.)
In [8] and [10], the authors give some conditions for a factor module of
an H-supplemented module to be H-supplemented. Now we give analogous
of their conditions for a τ -H-supplemented module.
120 H -supplemented modules
Theorem 1. Let τ be a cohereditary preradical. Let M be a τ -H-supp-
lemented module and X ≤ M . If for every direct summand K of M ,
(X+K)/X is a direct summand of M/X, then M/X is τ -H-supplemented.
Proof. Let N/X ≤ M/X. Since M is τ -H-supplemented, there exists a
direct summand D of M such that (N + D)/N ⊆ τ(M/N) and (N +
D)/D ⊆ τ(M/D). By assumption, (D + X)/X is a direct summand
of M/X. Since τ is a cohereditary preradical, it is easy to check that
N/X+(D+X)/X
N/X ⊆ τ(M/X
N/X ) and N/X+(D+X)/X
(D+X)/X ⊆ τ( M/X
(D+X)/X ). Hence M/X
is τ -H-supplemented.
Let M be a module. Then M is called distributive if its lattice of
submodules is a distributive lattice, equivalently for submodules K,L,N
of M , N+(K∩L) = (N+K)∩(N+L) or N∩(K+L) = (N∩K)+(N∩L).
Corollary 1. Let M be a τ -H-supplemented module for a cohereditary
preradical τ .
(1) If M is a distributive module, then M/X is τ -H-supplemented for
every submodule X of M .
(2) Let N ≤ M such that for each decomposition M = M1 ⊕M2 we have
N = (N ∩M1)⊕ (N ∩M2). Then M/N is τ -H-supplemented.
(3) Let X be a projection invariant submodule of M . Then M/X is τ -H-
supplemented. In particular, for every fully invariant submodule A of M ,
M/A is τ -H-supplemented.
Proof. (1) Let D be a direct summand of M . Then M = D⊕D′ for some
submodule D′ of M . Now M/X = [(D+X)/X]+ [(D′+X)/X] and X =
X+(D∩D′) = (X+D)∩(X+D′). So M/X = [(D+X)/X]⊕[(D′+X)/X].
By Theorem 1, M/X is τ -H-supplemented.
(2) Let L/N ≤ M/N . Since M is τ -H-supplemented, there exists a direct
summand D of M and a submodule X of M such that X/L ⊆ τ(M/L) and
X/D ⊆ τ(M/D). Let M = D⊕D′. Then by hypothesis, N = (D ∩N)⊕
(D′ ∩N) = (D +N) ∩ (D′ +N). So, (D +N)/N ⊕ (D′ +N)/N = M/N .
Now we have X/N
L/N ⊆ τ(M/N
L/N ) and X/N
(D+N)/N ⊆ τ( M/N
(D+N)/N ) and hence
M/N is τ -H-supplemented by Proposition 1.
(3) Clear by (2).
Proposition 2. Let M be a τ -H-supplemented module for a cohereditary
preradical τ and N ≤ M . If for each idempotent e : M → M there exists
an idempotent f : M/N → M/N such that (N+e(M))/N
T/N ⊆ τ(M/N
T/N ) where
Imf = T/N , then M/N is τ -H-supplemented.
Proof. Let Y/N ≤ M/N . Since M is τ -H-supplemented, there exists an
idempotent e : M → M and a submodule X of M such that X/e(M) ⊆
Y. Talebi, A. R. Moniri Hamzekolaei, D. Keskin 121
τ(M/e(M)) and X/Y ⊆ τ(M/Y ) by Proposition 1. By hypothesis, there
exists an idempotent f : M/N → M/N with Imf = T/N such that
(N + e(M))/T ⊆ τ(M/T ). Now, T/N is a direct summand of M/N and
T/N ⊆ X/N . Clearly X/N
T/N ⊆ τ(M/N
T/N ) and X/N
Y/N ⊆ τ(M/N
Y/N ).
Proposition 3. Let τ be a cohereditary preradical and M0 a direct sum-
mand of a module M such that for every decomposition M = N ⊕ K
of M , there exist submodules N ′ of N and K ′ of K such that M =
M0 ⊕N ′ ⊕K ′ with τ(K ′) = K ′. If M is τ -H-supplemented, then M/M0
is τ -H-supplemented.
Proof. Let N/M0 ≤ M/M0. Since M is τ -H-supplemented, there exists
a decomposition M = K ⊕ S such that (N + K)/N ⊆ τ(M/N) and
(N +K)/K ⊆ τ(M/K). By hypothesis, M = M0 ⊕N ′ ⊕K ′ for N ′ ≤ K
and K ′ ≤ S with τ(K ′) = K ′. Now it is easy to see that (M0 ⊕N ′)/M0
is a τ -H-supplement of N/M0 in M/M0.
Let M be an R-module and τ a preradical. By Pτ (M) we denote the
sum of all submodules N of M with τ(N) = N . The following Lemma
will be very useful for us to prove Corollary 2.
Lemma 1. Let τ be any preradical and let M be any module. Then
(1) τ(Pτ (M)) = Pτ (M).
(2) Pτ (M) is a fully invariant submodule of M .
(3) If M = N ⊕K, then Pτ (M) = Pτ (N)⊕ Pτ (K).
Corollary 2. Let M be a τ -H-supplemented module for a cohereditary pre-
radical τ . If Pτ (M) is a direct summand of M , then Pτ (M) and M/Pτ (M)
are τ -H-supplemented.
Proof. By Corollary 1(3) and Lemma 1(2), M/Pτ (M) is τ -H-suppleme-
nted. Let L be a submodule of M such that M = Pτ (M)⊕ L. Let M =
N⊕K. Now, by Lemma 1(3), M = Pτ (N)⊕Pτ (K)⊕L. Therefore M/L ∼=
Pτ (M) is τ -H-supplemented by Proposition 3 and Lemma 1(1).
2. Direct summands of τ-H-supplemented modules
In this section we will consider direct summands of τ -H-supplemented
modules. We investigate some conditions for direct summands of a τ -
H-supplemented module to be τ -H-supplemented. We call a module M
completely τ -H-supplemented provided every direct summand of M is τ -
H-supplemented. The following Theorem is an analogue of [10, Theorem
2.7].
122 H -supplemented modules
Theorem 2. (1) Every τ -lifting module is completely τ -H-supplemented.
(2)Let M be a τ -H-supplemented module for a cohereditary preradical τ .
If M has the SSP, then M is completely τ -H-supplemented.
Proof. (1) It is clear since by [2, 2.10] every direct summand of a τ -lifting
module is again τ -lifting.
(2) Assume that M is τ -H-supplemented and M has the SSP . Let N
be a direct summand of M . We will show that N is τ -H-supplemented.
Let M = N ⊕N ′ for some submodule N ′ of M . Suppose that A is a direct
summand of M . Since M has the SSP , A+N ′ is a direct summand of M .
Let M = (A+N ′)⊕B for some B ≤ M . Then M/N ′ = (A+N ′)/N ′ ⊕
(B +N ′)/N ′. Hence by Theorem 1, M/N ′ is τ -H-supplemented and so
N is τ -H-supplemented.
Proposition 4. Let M be a duo module. Then M has the SSP.
Proof. See [10, Page 969].
Corollary 3. Let τ be a cohereditary preradical. Let M be a τ -H-supple-
mented duo module. Then M is completely τ -H-supplemented.
The following is an example for Theorem 2(2) in case τ = Rad.
Example 2. Let F be a field and R the upper triangular matrix ring R =
(
F F
0 F
)
. Since R/JacR is semisimple, the preradical Rad is cohereditary.
For submodules A =
(
0 F
0 F
)
and B =
(
F F
0 0
)
, let M = A ⊕ (R/B).
Then M is H-supplemented by [6, Lemma 3]. Also M has the SSP .
Therefore M is a completely τ -H-supplemented module by Theorem 2(2).
3. Direct sums of τ-H-supplemented modules
The following example shows that any (finite) direct sum of τ -H-supplemen-
ted modules need not be τ -H-supplemented for τ = Rad. We will show
that under some conditions it will be true.
Example 3. Let R be a commutative local ring and M a finitely generated
R-module. Assume M ∼=
⊕n
i=1R/Ii. Since every Ii is fully invariant in
R, every R/Ii is τ -H-supplemented by Corollary 1(3). By [11, Lemma
A.4], M is τ -H-supplemented if I1 ≤ I2 ≤ ... ≤ In. If we don’t have the
condition I1 ≤ I2 ≤ ... ≤ In, M is not τ -H-supplemented. (Note that
since M is finitely generated, M is H-supplemented if and only if it is
τ -H-supplemented.)
Y. Talebi, A. R. Moniri Hamzekolaei, D. Keskin 123
We call a module M τ -semilocal provided that M/τ(M) is semisimple.
Clearly τ -supplemented modules are τ -semilocal.
Lemma 2. Let M be a τ -H-supplemented module for a cohereditary
preradical τ . Then M is τ -semilocal.
Proof. Let N/τ(M) ≤ M/τ(M). Since M is τ -H-supplemented, there
exists a direct summand D of M such that (N +D)/N ⊆ τ(M/N) and
(N +D)/D ⊆ τ(M/D). Since D ≤d M , M = D⊕D′ for some submodule
D′ of M . Then M = D′ + N . It follows that M/τ(M) = N/τ(M) +
(D′ + τ(M))/τ(M). Since N ∩D′ ⊆ τ(D′), M/τ(M) = N/τ(M)⊕ (D′ +
τ(M))/τ(M). Hence M/τ(M) is semisimple.
Proposition 5. Let M be a module. Then the following are equivalent
for a cohereditary preradical τ :
(1) M is τ -H-supplemented;
(2) M is τ -semilocal and each submodule (direct summand) of M/τ(M)
lifts to a direct summand of M .
Proof. (1) ⇒ (2) By Lemma 2, we only prove the last statement. Let
N/τ(M) ≤ M/τ(M). Since M is τ -H-supplemented, there exists D ≤d M
such that (N + D)/N ⊆ τ(M/N) and (N + D)/D ⊆ τ(M/D). Then
D ⊆ N . Hence N/τ(M) = (D + τ(M))/τ(M). This means N/τ(M) lifts
to D.
(2) ⇒ (1) Let N ≤ M . Then by assumption, (N + τ(M))/τ(M) = N
is a direct summand of M/τ(M) = M . Then by assumption N = L such
that M = L⊕K. The rest is easy by taking L as a τ -H-supplement of N
in M .
Theorem 3. Let τ be a cohereditary preradical. Let M = ⊕i∈IHi be a
direct sum of τ -H-supplemented modules Hi (i ∈ I). Assume that each
direct summand of M/τ(M) lifts to a direct summand of M . Then M is
τ -H-supplemented.
Proof. Clearly M/τ(M) is semisimple by Lemma 2. Now M is τ -H-
supplemented by Proposition 5.
Theorem 4. Let τ be a cohereditary preradical. Let M = M1 ⊕M2 be a
duo module. Then M is τ -H-supplemented if and only if M1 and M2 are
τ -H-supplemented.
Proof. Note that for A ≤ M , we can write A = (A ∩M1)⊕ (A ∩M2).
( ⇒ ) Assume that M is τ -H-supplemented. Since M1 and M2 are fully in-
variant submodules of M , M1 and M2 are τ -H-supplemented by Corollary
1(3).
124 H -supplemented modules
( ⇐ ) Suppose that M1 and M2 are τ -H-supplemented. Let A ≤ M . Then
A = (A∩M1)⊕ (A∩M2). By assumption, there exist direct summands D1
of M1 and D2 of M2 such that ((A∩M1)+D1)/(A∩M1) ⊆ τ(M1/(A∩M1)),
((A ∩ M1) + D1)/D1 ⊆ τ(M1/D1) and ((A ∩ M2) + D2)/(A ∩ M2) ⊆
τ(M2/(A∩M2)), ((A∩M2) +D2)/D2 ⊆ τ(M2/D2). It is not hard to see
that (A+ (D1 ⊕D2))/A ⊆ τ(M/A) and (A+ (D1 ⊕D2))/(D1 ⊕D2) ⊆
τ(M/(D1⊕D2)). Namely, D1⊕D2 is a τ -H-supplement of A in M . Hence
M is τ -H-supplemented.
Definition 2. Let M and N be two modules. Let τ be a preradical. Then
N is called τ -M -projective if, for any K ≤ M and any homomorphism
f : N −→ M/K there exists a homomorphism h : N −→ M such
that Im(f − πh) ⊆ τ(M/K), where π : M −→ M/K is the natural
epimorphism.
Lemma 3. Let M = M1 ⊕M2. Consider the following conditions:
1. M1 is τ -M2-projective;
2. For every K ≤ M with K +M2 = M , there exists M3 ≤ M such
that M = M2 ⊕M3 and (K +M3)/K ⊆ τ(M/K).
Then (1) ⇒ (2).
Proof. Let K ≤ M and M = K + M2. Consider the epimorphism π :
M2 −→ M/K with m2 7→ m2 + K(m2 ∈ M2) and the homomorphism
h : M1 −→ M/K with m1 7→ m1 + K(m1 ∈ M1). Since M1 is τ -M2-
projective, there exist a homomorphism h : M1 −→ M2 and a submodule
X of M with K ⊆ X such that Im(h − πh) = X/K ⊆ τ(M/K). Let
M3 = {a− h(a) | a ∈ M1}. Clearly M = M2 ⊕M3. Since K +M3 ⊆ X,
(K +M3)/K ⊆ X/K. Hence (K +M3)/K ⊆ τ(M/K).
Lemma 4. Let A and {Mi}
n
i=1 be modules. If each Mi is τ -A-projective,
for i = 1, 2, . . . n, then
⊕n
i=1Mi is τ -A-projective.
Proof. The proof is straightforward.
Theorem 5. Let τ be a cohereditary preradical. Let M = M1 ⊕ M2
be a τ -supplemented module. Assume M1 is τ -M2-projective (or M2 is
τ -M1-projective). If M1 and M2 are τ -H-supplemented, then M is τ -H-
supplemented.
Proof. Let Y ≤ M .
Case 1: Let M = Y +M2. Then by Lemma 3, there exists M3 ≤ M such
that M = M3 ⊕M2 and (Y +M3)/Y ⊆ τ(M/Y ). Since M/M3 is τ -H-
supplemented, there exist X/M3 ≤ M/M3 and a direct summand D/M3
Y. Talebi, A. R. Moniri Hamzekolaei, D. Keskin 125
of M/M3 such that X/M3
(Y+M3)/M3
⊆ τ( M/M3
(Y+M3)/M3
) and X/M3
D/M3
⊆ τ(M/M3
D/M3
)
by Proposition 1. Clearly, D is a direct summand of M . It is easy to
check that X/D ⊆ τ(M/D) and X/Y ⊆ τ(M/Y ). Therefore M is τ -H-
supplemented by Proposition 1.
Case 2: Let Y + M2 6= M . Since M is τ -supplemented, M/τ(M) is
semisimple. Then there exists a submodule K of M containing τ(M) such
that M/τ(M) = K/τ(M)⊕(Y +M2+τ(M))/τ(M). So M = (K+Y )+M2
and τ(M) = K ∩ (Y +M2 + τ(M)) = τ(M) + (K ∩ (Y +M2)) and hence
K ∩ (Y + M2) ⊆ τ(M). By Lemma 3, there exists M4 ≤ M such that
M = M2 ⊕ M4 and (K + Y + M4)/(K + Y ) ⊆ τ(M/(K + Y )). This
implies that K + Y +M4 ⊆ τ(M) +K + Y = K + Y . Now M/M2 and
M/M4 are τ -H-supplemented. Therefore there exist submodules X1/M2
of M/M2 and X2/M4 of M/M4 and direct summands D1/M2 of M/M2
and D2/M4 of M/M4 such that X1/M2
(Y+M2)/M2
⊆ τ( M/M2
(Y+M2)/M2
), X1/M2
D1/M2
⊆
τ( M/M2
D1/M2
), X2/M4
(Y+K+M4)/M4
⊆ τ( M/M4
(Y+K+M4)/M4
) and X2/M4
D2/M4
⊆ τ( M/M4
D2/M4
).
Clearly,D1∩D2 is a direct summand of M . Let M = (D1∩D2)⊕L for some
submodule L of M . Then by [7, Lemma 1.2], M = D2⊕(D1∩L). Note that
we have that X1 ⊆ τ(M) +D1, X1 ⊆ τ(M) +M2 + Y , X2 ⊆ τ(M) +D2
and X2 ⊆ τ(M) + Y +K +M4 = K + Y . Now,
X1 ∩X2 ⊆ (τ(M) +M2 + Y ) ∩ (Y +K)
= (τ(M) + Y ) + (M2 ∩ (Y +K))
⊆ τ(M) + Y + [K ∩ (Y +M2)] + [Y ∩ (K +M2)]
= τ(M) + Y
and
X1 ∩X2 ⊆ (τ(M) +D1) ∩ (τ(M) +D2)
= (τ(D2) +D1) ∩ (τ(D1 ∩ L) +D2)
= τ(D2) + [(D2 + τ(D1 ∩ L)) ∩D1]
= τ(D2) + τ(D1 ∩ L) + (D1 ∩D2)
⊆ τ(M) + (D1 ∩D2).
Therefore (X1∩X2)/Y ⊆ τ(M/Y ) and (X1∩X2)/(D1∩D2) ⊆ τ(M/(D1∩
D2)). Thus M is τ -H-supplemented by Proposition 1.
Corollary 4. Let τ be a cohereditary preradical. Let M = ⊕n
i=1Mi be a
τ -supplemented module. Assume that Mi is τ -Mj-projective for all j > i.
If each Mi is τ -H-supplemented, then M is τ -H-supplemented.
Proof. By Lemma 4 and Theorem 5.
126 H -supplemented modules
4. Relations between τ-H-supplemented modules and the
others
A module M is called τ -⊕-supplemented if for every A ≤ M , there exists
a B ≤d M such that A + B = M and A ∩ B ⊆ τ(B). Clearly every τ -
lifting module is τ -⊕-supplemented and every τ -⊕-supplemented module
is τ -supplemented.
Next we will show that under some conditions every τ -⊕-supplem-
ented module is τ -H-supplemented.
Proposition 6. Let τ be any preradical. Assume M is τ -⊕-supplemented
such that whenever M = M1⊕M2 then M1 and M2 are relatively projective.
Then M is τ -H-supplemented.
Proof. Let N ≤ M . Since M is τ -⊕-supplemented, there exists a decom-
position M = M1 ⊕M2 such that M = N +M2 and N ∩M2 ⊆ τ(M2)
for submodules M1,M2 of M . By hypothesis, M1 is M2-projective. By
[11, Lemma 4.47], we obtain M = A ⊕ M2 for some submodule A of
M such that A ≤ N . Then N = A ⊕ (M2 ∩ N). It is easy to see that
(N + A)/A ⊆ τ(M/A) and (N + A)/N ⊆ τ(M/N). Thus M is τ -H-
supplemented.
Corollary 5. Let τ be any preradical. Let M be a τ -⊕-supplemented
module. If M is projective, then M is τ -H-supplemented.
Let e = e2 ∈ R. Then e is called a left (right) semicentral idempotent
if xe = exe (ex = exe), for all x ∈ R. The set of all left (right) semicentral
idempotents is denoted by Sl(R) (Sr(R)). A ring R is called Abelian if
every idempotent is central.
Proposition 7. Let τ be a preradical and M an R-module such that
End(M) is Abelian and X ≤ M implies X =
∑
i∈I hi(M) where hi ∈
End(M). If M is τ -⊕-supplemented, then M is τ -H-supplemented and
satisfies the (D3)-condition.
Proof. Let X ≤ M , X =
∑
i∈I hi(M) with hi ∈ End(M). Since M is τ -⊕-
supplemented, there exists a direct summand eM such that X + eM = M
and (X ∩ eM) ⊆ τ(eM) for some e2 = e ∈ End(M). Since End(M) is
Abelian, (1−e)X = (1−e)M = (1−e)
∑
i∈I hi(M) =
∑
i∈I hi(1−e)(M) ⊆
X. Therefore X = (1 − e)M ⊕ (X ∩ eM). Then (1 − e)M is a τ -H-
supplement of X. If eM + fM = M for e2 = e, f2 = f ∈ End(M), then
eM ∩fM = efM with (ef)2 = ef . So M satisfies the (D3)-condition.
Recall that for a commutative ring R, an R-module M is said to be a
multiplication module if for each X ≤ M , X = MA for some ideal A of R.
Y. Talebi, A. R. Moniri Hamzekolaei, D. Keskin 127
Corollary 6. Let τ be a preradical and M a τ -⊕-supplemented module. If
M satisfies one of the following conditions, then M is τ -H-supplemented.
(1) M is a multiplication module and R is commutative.
(2) M is cyclic and R is commutative.
Proof. (1) Assume M is a multiplication module. Let X ≤ M . Then
X = MA for some ideal A of R. For each a ∈ A, define ha : M → M
by ha(m) = ma for all m ∈ M . Then ha is an R-homomorphism and
X = MA =
∑
a∈A ha(M). Since every multiplication module is a duo
module, thus if e2 = e ∈ S = End(M), then e, 1−e ∈ Sl(S). Therefore e is
central. So End(M) is Abelian. By Proposition 7, M is τ -H-supplemented.
(2) Clear by (1) since every cyclic module over a commutative ring is a
multiplication module.
Now we investigate the relations between τ -H-supplemented modules
and the others. A module M is called amply τ -supplemented if for any
submodules K and V of M such that M = K + V , there is a submodule
U of V such that K + U = M and K ∩ U ⊆ τ(U).
Lemma 5. Let τ be any preradical and let M be a projective module. The
following are equivalent:
(1) M is τ -supplemented;
(2) M is amply τ -supplemented.
Proof. Clearly an amply τ -supplemented module is τ -supplemented. For
the converse: Let M = U + V and X be a τ -supplement of U in M . For
an f ∈ End(M) with Im(f) ⊆ V and Im(I − f) ⊆ U we have f(U) ⊆ U ,
M = U + f(X) and f(U ∩ X) = U ∩ f(X) (from u = f(x) we derive
x− u = (I − f)(x) ∈ U and x ∈ U). Since U ∩X ⊆ τ(X), we also have
U ∩ f(X) ⊆ τ(f(X)), i.e. f(X) is a τ -supplement of U with f(X) ⊆ V .
Hence M is amply τ -supplemented.
Let M be any module. A submodule U of M is called quasi strongly
lifting (QSL) in M if whenever (A+ U)/U is a direct summand of M/U ,
there exists a direct summand P of M such that P ≤ A and P+U = A+U
(see [1]).
Lemma 6. Let τ be a cohereditary preradical and let M be any module.
The following are equivalent:
(1) M is τ -lifting;
(2) M is τ -H-supplemented and τ(M) is QSL in M .
Proof. By Lemma 2 and [1, Lemma 3.5 and Proposition 3.6].
128 H -supplemented modules
Lemma 7. Let τ be any preradical and let M be a projective module such
that every τ -supplement submodule of M is a direct summand of M . The
following are equivalent:
(1) M is τ -supplemented;
(2) M is amply τ -supplemented;
(3) M is τ -lifting;
(4) M is τ -⊕-supplemented.
Proof. (1) ⇔ (2) By Lemma 5.
(1) ⇒ (3) By [1, Lemma 3.2].
(3) ⇒ (1) and (1) ⇔ (4) are clear by definitions and the assumption
that every τ -supplement submodule of M is a direct summand of M .
Now we have the following Theorem:
Theorem 6. Let τ be a cohereditary preradical. Let M be a projective
module such that every τ -supplement submodule of M is a direct summand.
The following are equivalent:
(1) M is τ -supplemented;
(2) M is τ -lifting;
(3) M is amply τ -supplemented;
(4) M is τ -H-supplemented and τ(M) is QSL in M ;
(5) M is τ -⊕-supplemented.
As we see in Example 3 a finite direct sum of τ -H-supplemented
modules need not be τ -H-supplemented. We will show that a finite direct
sum of τ -⊕-supplemented modules is τ -⊕-supplemented.
Lemma 8. Let N,L ≤ M such that N + L has a τ -supplement H in
M and N ∩ (H + L) has a τ -supplement G in N . Then H + G is a
τ -supplement of L in M .
Proof. Let H be a τ -supplement of N+L in M and G be a τ -supplement of
N ∩ (H+L) in N . Then M = (N +L)+H such that (N +L)∩H ⊆ τ(H)
and N = [N ∩ (H + L)] + G such that (H + L) ∩ G ⊆ τ(G). Since
(H +G)∩L ⊆ [(G+L)∩H ] + [(H +L)∩G] ⊆ τ(H)+ τ(G) ⊆ τ(H +G),
H +G is a τ -supplement of L in M .
Theorem 7. For a ring R, any finite direct sum of τ -⊕-supplemented
R-modules is τ -⊕-supplemented.
Proof. Let M = M1 ⊕ ...⊕Mn and Mi be a τ -⊕-supplemented module
for each 1 ≤ i ≤ n. To prove that M is τ -⊕-supplemented it is sufficient
to assume n = 2.
Y. Talebi, A. R. Moniri Hamzekolaei, D. Keskin 129
Let L ≤ M . Then M = M1 + M2 + L so that M1 + M2 + L has a
τ -supplement 0 in M . Let H be a τ -supplement of M2 ∩ (M1 + L) in M2
such that H ≤d M2. By Lemma 8, H is a τ -supplement of M1 + L in M .
Let K be a τ -supplement of M1 ∩ (L + H) in M1 such that K ≤d M1.
Again by applying Lemma 8, we get that H +K is a τ -supplement of L in
M . Since H ≤d M2 and K ≤d M1, it follows that H +K = H ⊕K ≤d M .
Thus M = M1 ⊕M2 is τ -⊕-supplemented.
Note that by the same proof as the proof of Theorem 7, any finite sum
of τ -supplemented modules is τ -supplemented.
Theorem 8. Let τ be a cohereditary preradical. Let R be a τ -⊕-supple-
mented ring (i.e. RR is τ -⊕-supplemented) such that every finite direct
sum of the copies of R is distributive. Then the following are equivalent:
(1) R is τ -H-supplemented;
(2) Every finitely generated free R-module is τ -H-supplemented;
(3) Every finitely generated projective R-module is τ -H-supplemented;
(4) If F is a finitely generated free R-module and N a fully invariant
submodule, then F/N is τ -H-supplemented.
Proof. (1) ⇒ (3) Let M be a finitely generated projective R-module. Then
M is isomorphic to a direct summand of a finitely generated free module
F . By Corollary 4, F is τ -H-supplemented. Thus M is τ -H-supplemented
by Corollary 1(1).
(3) ⇒ (2) ⇒ (1) and (4) ⇒ (1) are clear.
(2) ⇒ (4) By (2), F is τ -H-supplemented. The result follows from
Corollary 1(3).
We next consider the preradical Z.
Let M be a module and S denote the class of all small modules. Talebi
and Vanaja defined Z(M) in [13] as follows:
Z(M) =
⋂
{kerg | g ∈ Hom(M,L), L ∈ S}. The module M is called
cosingular (non-cosingular) if Z(M) = 0 (Z(M) = M). Clearly every
non-cosingular module is Z-H-supplemented. Also if R is a non-cosingular
ring, then every R-module is Z-H-supplemented by [13, Proposition 2.5
and Corollary 2.6].
Let M be a module and τM a preradical on σ[M ]. In [12], the authors
call a module N ∈ σ[M ] τM -semiperfect if it satisfies one of the following
conditions (see [12, Proposition 2.1 and Definition 2.2]):
(1) For every submodule K of N there exists a decomposition K =
A ⊕ B such that A is a projective direct summand of N in σ[M ] and
B ⊆ τM (N);
130 H -supplemented modules
(2) For every submodule K of N , there exists a decomposition N =
A⊕B such that A is projective in σ[M ], A ≤ K and K ∩B ⊆ τM (N).
If σ[M ] = Mod− R, then they call N τ -semiperfect.
By the above definition, every τ -semiperfect module is τ -lifting and
hence τ -H-supplemented. Also if M is projective we have the following:
τ -semiperfect ⇔ τ -lifting ⇔ τ -⊕-supplemented ⇒ τ -H-supplemented
In [12, Theorem 2.23], the authors showed that their τ -semiperfect
module definition agrees with the definition of τ -semiperfect module in
the sense of [2] for a projective module and for the preradical Soc. In [14],
Tribak and Keskin Tütüncü studied Z-lifting modules and Z-semiperfect
modules in the sense of [12]. They also investigate some conditions for
the preradical Z for two definitions of τ -semiperfect modules to be agreed
(see [14, Proposition 5.8 and Proposition 5.11]).
A τ -H-supplemented module need not be H-supplemented. Of course
if τ(M) ≪ M and τ is cohereditary, then every τ -H-supplemented module
is H-supplemented.
Example 4. Let K be a field and let R =
∏
n≥1Kn with Kn = K. By
[14, Example 4.1(1)] R is not semiperfect. Since R is projective, R is
not ⊕-supplemented by [5, Lemma 1.2]. Hence R is not H-supplemented.
Again by [14, Example 4.1(1)], the module R is Z-semiperfect in the sense
of [12] and so it is Z-H-supplemented.
If R is a DV R (Discrete Valuation Ring), then the R-module R is
semiperfect and hence H-supplemented.
Now we give an equivalent condition for a module to be Z-⊕-supplem-
ented module under some assumptions.
Proposition 8. Let R be a commutative ring, P a projective module with
Rad(P ) ≪ P and assume P to have finite hollow dimension. Then the
following are equivalent:
(1) P is Z-⊕-supplemented;
(2) P = P1 ⊕ P2 ⊕ P3 with P1 ⊕-supplemented and Rad(P1) = Z(P1),
P2 semisimple and Z(P3) = P3.
Proof. (1) ⇒ (2) See the proof of [14, Corollary 4.3] and [5, Lemma 2.1].
(2) ⇒ (1) By [14, Corollary 4.3] all P1, P2 and P3 are Z-semiperfect
in the sense of [12] and hence Z-⊕-supplemented. By Theorem 7, P is
Z-⊕-supplemented.
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Y. Talebi, A. R. Moniri Hamzekolaei, D. Keskin 131
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Contact information
Yahya Talebi,
Ali Reza Moniri
Hamzekolaei
Department of Mathematics, Faculty of Basic Sci-
ence, University of Mazandaran, Babolsar, Iran
E-Mail: talebi@umz.ac.ir,
a.monirih@umz.ac.ir
Derya Keskin
Tütüncü
Department of Mathematics, Hacettepe Univer-
sity, 06800 Beytepe, Ankara, Turkey
E-Mail: keskin@hacettepe.edu.tr
Received by the editors: 14.11.2009
and in final form 01.10.2011.
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