On generalization of ⊕-cofinitely supplemented modules
We study the properties of ⊕-cofinitely radical supplemented modules, or, briefly, cgs ⊕-modules. It is shown that a module with summand sum property (SSP) is cgs ⊕ if and only if M/w Loc⊕ M (w Loc⊕ M is the sum of all w-local direct summands of a module M) does not contain any maximal submodule, th...
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irk-123456789-1646522020-02-11T01:26:05Z On generalization of ⊕-cofinitely supplemented modules Nisanci, B. Pancar, A. Статті We study the properties of ⊕-cofinitely radical supplemented modules, or, briefly, cgs ⊕-modules. It is shown that a module with summand sum property (SSP) is cgs ⊕ if and only if M/w Loc⊕ M (w Loc⊕ M is the sum of all w-local direct summands of a module M) does not contain any maximal submodule, that every cofinite direct summand of a UC-extending cgs ⊕-module is cgs ⊕, and that, for any ring R, every free R-module is cgs ⊕ if and only if R is semiperfect. Досліджено властивості ⊕-кофінітно радикальних поповнених модулів або скорочено cgs ⊕-модулів. Показано, що модуль із властивістю суми доданків SSP є cgs⊕-модулем тоді і тільки тоді, колиM/wLoc⊕M (wLoc⊕M — сума всіх w-локальних прямих доданків модуля M) не містить жодного максимального субмодуля; кожний прямий доданок UC-розширюваного cgs⊕-модуля є cgs⊕-модулем; для будь-якого кільця R кожний вільний R-модуль є cgs⊕-модулем тоді і тільки тоді, коли R є напівперфектним. 2010 Article On generalization of ⊕-cofinitely supplemented modules / B. Nisanci, A. Pancar // Український математичний журнал. — 2010. — Т. 62, № 2. — С. 183–189. — Бібліогр.: 12 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164652 512.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Nisanci, B. Pancar, A. On generalization of ⊕-cofinitely supplemented modules Український математичний журнал |
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We study the properties of ⊕-cofinitely radical supplemented modules, or, briefly, cgs ⊕-modules. It is shown that a module with summand sum property (SSP) is cgs ⊕ if and only if M/w Loc⊕ M (w Loc⊕ M is the sum of all w-local direct summands of a module M) does not contain any maximal submodule, that every cofinite direct summand of a UC-extending cgs ⊕-module is cgs ⊕, and that, for any ring R, every free R-module is cgs ⊕ if and only if R is semiperfect. |
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Nisanci, B. Pancar, A. |
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Nisanci, B. Pancar, A. |
| author_sort |
Nisanci, B. |
| title |
On generalization of ⊕-cofinitely supplemented modules |
| title_short |
On generalization of ⊕-cofinitely supplemented modules |
| title_full |
On generalization of ⊕-cofinitely supplemented modules |
| title_fullStr |
On generalization of ⊕-cofinitely supplemented modules |
| title_full_unstemmed |
On generalization of ⊕-cofinitely supplemented modules |
| title_sort |
on generalization of ⊕-cofinitely supplemented modules |
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Інститут математики НАН України |
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2010 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/164652 |
| citation_txt |
On generalization of ⊕-cofinitely supplemented modules / B. Nisanci, A. Pancar // Український математичний журнал. — 2010. — Т. 62, № 2. — С. 183–189. — Бібліогр.: 12 назв. — англ. |
| series |
Український математичний журнал |
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AT nisancib ongeneralizationofcofinitelysupplementedmodules AT pancara ongeneralizationofcofinitelysupplementedmodules |
| first_indexed |
2025-07-14T17:15:48Z |
| last_indexed |
2025-07-14T17:15:48Z |
| _version_ |
1837643437826899968 |
| fulltext |
UDC 512.5
B. Nisanci, A. Pancar (Ondokuz Mayıs Univ., Turkey)
ON GENERALIZATION OF ⊕-COFINITELY
SUPPLEMENTED MODULES
ПРО УЗАГАЛЬНЕННЯ ⊕-КОФIНIТНО
ПОПОВНЕНИХ МОДУЛIВ
We study the properties of ⊕-cofinitely radical supplemented modules or briefly cgs⊕-modules. It is shown
that: a module with Summand Sum Property (SSP) is cgs⊕ if and only if M/w Loc⊕M (w Loc⊕M is the
sum of all w-local direct summands of a module M) does not contain any maximal submodule; every cofinite
direct summand of a UC-extending cgs⊕-module is cgs⊕; for any ring R, every free R-module is cgs⊕ if
and only if R is semiperfect.
Дослiджено властивостi ⊕-кофiнiтно радикальних поповнених модулiв або скорочено cgs⊕-модулiв.
Показано, що модуль iз властивiстю суми доданкiв SSP є cgs⊕-модулем тодi i тiльки тодi, коли
M/w Loc⊕M (w Loc⊕M — сума всiх w-локальних прямих доданкiв модуля M) не мiстить жодного
максимального субмодуля; кожний прямий доданок UC-розширюваного cgs⊕-модуля є cgs⊕-модулем;
для будь-якого кiльця R кожний вiльний R-модуль є cgs⊕-модулем тодi i тiльки тодi, коли R є напiв-
перфектним.
1. Introduction. In this note R will be an associative ring with identity and all modules
are unital left R-modules. Let M be an R-module. The notation N ⊆ M means that N
is a submodule of M. RadM will indicate Jacobson radical of M. A submodule N of an
R-module M is called small in M (notation N �M), if N + L 6= M for every proper
submodule L of M. Let M be an R-module and let N and K be any submodules of
M. K is called a supplement of N in M if M = N + K and N ∩K � K (see [1]).
Following [1], M is called supplemented if every submodule of M has a supplement
in M. A submodule N of a module M is called cofinite in M if the factor module
M
N
is finitely generated. A module M is called cofinitely supplemented if every cofinite
submodule of M has a supplement in M (see [2]). Clearly supplemented modules are
cofinitely supplemented. A module M is called ⊕-supplemented if every submodule of
M has a supplement that is a direct summand of M (see [3]). As a proper generalization
of ⊕-supplemented modules, the notation of ⊕-cofinitely supplemented modules was
introduced by Calisici and Pancar [4]. A module M is called ⊕-cofinitely supplemented
if every cofinite submodule of M has a supplement that is a direct summand of M. Also,
finitely generated ⊕-cofinitely supplemented modules are ⊕-supplemented.
In [5] (Theorem 10.14), another generalization of supplement submodule was called
as radical supplement or briefly Rad-supplement (according to [6], generalized supp-
lement). For a module M and a submodule N of M, a submodule K of M is called
a Rad-supplement of N in M if N + K = M and N ∩ K ⊆ RadK. An R-module
M is called radical supplemented or briefly Rad-supplemented if every submodule of
M has a Rad-supplement in M (in [6], generalized supplemented or GS-module). Since
the Jacobson radical of a module is sum of all small submodules, every supplement is
a Rad-supplement. Therefore every supplemented module is Rad-supplemented. In [7],
M is called cofinitely radical supplemented or briefly cofinitely Rad-supplemented if
every cofinite submodule of M has a Rad-supplement in M. Clearly Rad-supplemented
modules are cofinitely Rad-supplemented.
c© B. NISANCI, A. PANCAR, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 2 183
184 B. NISANCI, A. PANCAR
Let M be an R-module. M is called ⊕-radical supplemented or briefly ⊕-Rad-
supplemented or generalized ⊕ -supplemented if every submodule of M has a Rad-
supplement that is a direct summand of M. Clearly ⊕-Rad-supplemented modules are
Rad-supplemented. A module M is called ⊕-cofinitely radical supplemented (according
to [8], generalized ⊕-cofinitely supplemented) if every cofinite submodule of M has
a Rad-supplement that is a direct summand of M. Instead of a ⊕-cofinitely radical
supplemented module, we will use a cgs⊕-module.
In this paper we study the properties of cgs⊕-modules as both a proper generalization
of ⊕-Rad-supplemented modules and a generalization of ⊕-cofinitely supplemented
modules. We prove that a module M with SSP is cgs⊕ if and only if M/w Loc⊕ M
does not contain any maximal submodule, where w Loc⊕ M is the sum of all w-local
direct summands of M. Also we show that any direct sum of cgs⊕-modules is also a
cgs⊕-module. Using the mentioned fact we give a characterization of semiperfect rings.
2. Some properties of ⊕-cofinitely radical supplemented modules. It is clear
that every ⊕-cofinitely supplemented module is cgs⊕, but it is not generally true that
every cgs⊕-module is ⊕-cofinitely supplemented. Later we shall give an example of
such modules (see Example 2.1). Now we give an analogue of these modules.
Proposition 2.1. Let M be a cgs⊕-module with small radical. Then M is ⊕-
cofinitely supplemented.
Proof. Let U be any cofinite submodule of M. By the hypothesis, there exist
submodules V, V ′ of M such that M = U + V, U ∩ V ⊆ RadV and M = V ⊕ V ′.
Since U ∩ V ⊆ RadV ⊆ RadM � M and V is a direct summand of M, then
U ∩ V � V by [1] (19.3.(5)). Hence M is ⊕-cofinitely supplemented.
Let M be an R-module. If every proper submodule of M is contained a maximal
submodule of M, M is called coatomic. Note that every coatomic module has small
radical.
Corollary 2.1. Let M be a coatomicR-module. Then M is a cgs⊕-module if and
only if it is ⊕-cofinitely supplemented.
Every cgs⊕-module is cofinitely Rad-supplemented but the converse is not true.
For example, a left (cofinitely) Rad-supplemented ring which is not supplemented (i.e.,
semiperfect) is cofinitely Rad-supplemented over itself, but not a cgs⊕-module.
Therefore we have the following implications on modules:
⊕-supplemented
↙ ↘
⊕-cofinitely supplemented ⊕- Rad-supplemented
↘ ↙ ↘
⊕-cofinitely radical supplemented Rad-supplemented
↘ ↙
cofinitely Rad-supplemented
We begin by some general properties of cgs⊕-modules. To prove that any direct sum
of cgs⊕-modules is cgs⊕, we use the following standart Lemma ([7], 3.4).
Lemma 2.1. Let M be an R-module and N, U be submodules of M such that N
is cofinitely Rad-supplemented, U cofinite and N + U has a Rad -supplement A in M.
Then N ∩ (U + A) has a Rad -supplement B in N and A + B is a Rad-supplement of
U in M.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 2
ON GENERALIZATION OF ⊕-COFINITELY SUPPLEMENTED MODULES 185
Proof. Let A be a Rad-supplement of N + U in M. Then
N
N ∩ (U + A)
∼=
N + U + A
U + A
∼=
M/U
(U + A) /U
.
Since U is a cofinite submodule of N, N ∩ (U + A) is cofinite. By hypothesis, N
is cofinitely Rad-supplemented, N ∩ (U + A) has a Rad-supplement B in N. Then
M = (N + U) + A = U + A + B and by [1] (19.3), U ∩ (A + B) ⊆ A ∩ (U + B) +
+ B ∩ (U + A) ⊆ A ∩ (N + U) + B ∩ (U + A) ⊆ Rad (A + B). Therefore A + B is
a Rad-supplement of U in M.
Theorem 2.1. For any ring R, any direct sum of cgs⊕-modules is a cgs⊕-module.
Proof. Let R be any ring and {Mi}i∈I be any family of cgs⊕-modules. Let M =
= ⊕i∈IMi and N be a cofinite submodule of M. Then M = ⊕n
j=1Mij
+ N and it
is clear that {0} is Rad-supplement of M = Mi1 + (⊕n
j=2Mij
+ N). Since Mi1 is a
cgs⊕-module, Mi1∩(⊕n
j=2Mij
+N) has a Rad-supplement Vi1 in Mi1 such that Vi1 is a
direct summand of Mi1. By Lemma 2.1, Vi1 is a Rad-supplement of⊕n
j=2Mij +N in M.
Note that since Mi1 is a direct summand of M, Vi1 is also a direct summand of M. By
repeated use of Lemma 2.1, since the set J is finite at the end we will obtain that N has
a Rad-supplement Vi1 +Vi2 + . . .+Vir
in M such that every Vij
, 1 ≤ j ≤ n, is a direct
summand of Mij
. Since every Mij
is a direct summand of M,
∑n
j=1
Vij
= ⊕n
j=1Vij
is a direct summand of M. Hence M is a cgs⊕-module.
Recall from [7] that a module M is called w-local if it has a unique maximal
submodule. It is clear that a module is w-local if and only if its radical is maximal.
Local modules are w-local. But it is not generally true that every w-local module is
local. For example, p any prime, the Z-module Q⊕Zp is w-local but it is not local. It is
trivial that w-local modules are a generalization of local modules. This fact plays a key
role in our working.
Proposition 2.2. The following statements are equivalent for a w-local module M.
(i) RadM �M.
(ii) M is finitely generated.
Proof. Suppose that M is a w-local module. Then RadM is a maximal submodule
of M. Thus RadM + Rm = M for every m ∈ M \RadM. Since RadM � M, then
Rm = M. Hence M is finitely generated. The converse is clear.
Proposition 2.3. Let M be a w-local R-module. Then M is a cgs⊕-module.
Proof. It follows from [7] (Lemma 3.2).
Proposition 2.4. Let M be a cgs⊕-module. If M has a maximal submodule, then
M contains a w-local direct summand.
Proof. Let L be a maximal submodule of M. Then L is cofinite and it follows
that there exist K, K ′ submodules of M such that L + K = M, L ∩ K ⊆ RadK
and M = K ⊕ K ′. By Lemma 3.3 in [7], K is w-local. Hence K is a w-local direct
summand of M.
Let M be an R-module. w Loc⊕ M will denote the sum of all w-local direct
summands of M.
Recall from [1] that an R-module M has Summand Sum Property (SSP) if the sum
of two direct summands of M is again a direct summand of M.
We give a characterization of cgs⊕-modules. Firstly we need the following lemma
which is a generalization of [2] (Lemma 2.9).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 2
186 B. NISANCI, A. PANCAR
Lemma 2.2. Let M be an R-module and N be a cofinite submodule of M. Let
{Li}n
i=1 be the family of w-local submodules such that K is a Rad-supplement of
N + L1 + . . . + Ln in M. Then K +
∑
i∈I
Li is a Rad-supplement of N in M such
that I is a subset of {1, 2, . . . , n}.
Proof. Suppose that n = 1. Consider the submodule H = (N + K) ∩ L1 of L1. K
is a Rad-supplement of N +L1, so that M = N +L1 +K and (N +L1)∩K ⊆ RadK.
Then H is a cofinite submodule of L1. Since L1 is w-local, then RadL1 is a unique
maximal submodule of L1. Note that H ⊆ RadL1. By [9] (19.3), N ∩ (K + L1) ⊆
⊆ K ∩ (N + L1) + H ⊆ RadK + RadL1 ⊆ Rad(K + L1). Therefore K + L1 is
a Rad-supplement of N. This proves the result when n = 1. Suppose that n ≥ 2. By
induction on n, there exists a subset I ′ of {2, 3, . . . , n} such that K +
∑
i∈I′
Li is a
Rad-supplement of N + L1 in M. Now the case n = 1 shows that K + L1 +
∑
i∈I′
Li
is a Rad-supplement of N in M.
Theorem 2.2. Let R be any ring and M be an R-module with SSP. Then the
following statements are equivalent.
(i) M is a cgs⊕-module.
(ii) Every maximal submodule of M has a Rad-supplement that is a direct summand
of M.
(iii) M/w Loc⊕ M does not contain a maximal submodule.
Proof. (i) ⇒ (ii) Clear.
(ii)⇒ (iii). Suppose that M/w Loc⊕ M contains a maximal submodule U/w Loc⊕ M.
Then U is a maximal submodule of M. By assumption, U has a Rad-supplement V that
is a direct summand of M. Then V is w-local and it follows that V ⊆ w Loc⊕ M. Since
M = U + V and w Loc⊕ M ⊆ U, we get M = U which is a contradiction.
(iii)⇒ (i). Let N be any cofinite submodule of M. Then N +w Loc⊕ M is a cofinite
submodule of M. By (iii), M = N + w Loc⊕ M. Because M/N is finitely generated,
there exist w-local submodules Li, 1 ≤ i ≤ n, for some positive integer n, such that
each of them is a direct summand of M and M = N +
∑n
i=1
Li has a Rad-supplement
{0} in M. By Lemma 2.2,
∑
i∈I′
Li is a Rad-supplement of N in M such that I ′ is
a subset of {1, 2, . . . , n}. Moreover
∑
i∈I′
Li is a direct summand of M. Thus M is a
cgs⊕-module.
Example 2.1. Let R be a commutative local ring which is not a valuation ring.
Let x and y be elements of R, neither of them divides the other. By taking a suitable
quotient ring, we may assume that (x) ∩ (y) = 0 and xI = yI = 0, where I is the
maximal ideal of R. Let F be a free module with generators a1, a2, a3. Let N be the
submodule generated by xa1−ya2 and M = F/N. R is local, so RR is a cgs⊕-module.
By Theorem 2.1, F is a cgs⊕-module. Suppose that M is a cgs⊕-module. Since F is
finitely generated, M is finitely generated and it follows that M has a small radical.
By Proposition 2.1, M is ⊕-(cofinitely) supplemented. This is a contradiction by [10]
(Example 2.3).
This example shows that the factor module of a cgs⊕-module is not in general cgs⊕.
Let R be a ring and M be an R-module. We consider the following condition.
(D3) If K and N are direct summands of M with M = K + N, then K ∩N is also
a direct summand of M (see [11]).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 2
ON GENERALIZATION OF ⊕-COFINITELY SUPPLEMENTED MODULES 187
Proposition 2.5. Let M be a cgs⊕-module with (D3) . Then every cofinite direct
summand of M is a cgs⊕-module.
Proof. Let N be any cofinite direct summand of M. Then there exists a submodule
N ′ of M such that M = N ⊕ N ′ and N ′ is finitely generated. Let U be any cofinite
submodule of N. Note that M/U ∼= N/U ⊕ N ′ is finitely generated so that U is also
cofinite submodule of M. Since M is a cgs⊕-module, then there exists a direct summand
V of M such that M = U + V and U ∩ V ⊆ RadV. Hence N = U + (N ∩ V ). Since
M has (D3), N ∩V is a direct summand of M. Furthermore N ∩V is a direct summand
of N because N is a direct summand of M. Then U ∩ (N ∩ V ) = U ∩ V ⊆ RadM.
Note that U ∩ (N ∩ V ) ⊆ Rad (N ∩ V ) by [1] (19.3). Hence N is a cgs⊕-module.
Corollary 2.2. Let M be a UC-extending module. If M is a cgs⊕-module, then
every cofinite direct summand of M is a cgs⊕-module.
Recall from [1] that a submodule U of an R-module M is called fully invariant if
f (U) is contained in U for every R-endomorphism f of M. Let M be an R-module
and τ be a preradical for the category of R-modules. Then, RadM and τ (M) are fully
invariant submodule of M. An R-module M is called a (weak) duo module if every
(direct summand) submodule of M is fully invariant. Note that weak duo modules has
SSP (see [9]).
Corollary 2.3. Let R be a ring and M be a weak duo R-module. Then M is a
cgs⊕-module if and only if every maximal submodule of M has a Rad-supplement that
is a direct summand of M.
Proposition 2.6. Let M be a cgs⊕-module and U be a fully invariant submodule
of M. Then M/U is a cgs⊕-module.
Proof. Let K/U be a cofinite submodule of M/U. Then K is a cofinite submodule
of M. Since M is a cgs⊕-module, then (N + U) /U is a Rad-supplement of K/U in
M/U by [6] (Proposition 2.6) and M = N ⊕ N ′ for N ′ is a submodule of M. By
hypothesis, U is a fully invariant submodule of M. Note that U = (U ∩N)⊕ (U ∩N ′)
by [9] (Lemma 2.1). Then M/U = (N + U) /U ⊕ (N ′ + U) /U. (N + U) /U is a
Rad-supplement of K/U such that (N + U) /U is a direct summand of M/U. Hence
M/U is a cgs⊕-module.
Corollary 2.4. Let M be a cgs⊕-module. Then M/RadM and M/τ (M) is a
cgs⊕-module.
Proposition 2.7. Let M be a cgs⊕-module and U be a fully invariant submodule
of M. If U is a cofinite direct summand of M, then U is a cgs⊕-module.
Proof. Let U be a cofinite submodule of M. Since U is a cofinite direct summand
of M, it follows that U ⊕ U ′ = M for U ′ ⊆ M. Let V be a cofinite submodule
of U. Then U/V and U ′ is finitely generated. Therefore V is a cofinite submodule
of M. By hypothesis, V + K = M, V ∩ K ⊆ RadK and M = K ⊕ K ′ such
that K, K ′ ⊆ M. Note that U = (U ∩K) ⊕ (U ∩K ′) by [9] (Lemma 2.1). Then
U = V ⊕ (U ∩K) and V ∩ (U ∩K) ⊆ RadM. Since U ∩K is a direct summand of
M, then V ∩ (U ∩K) ⊆ Rad(U ∩K). U ∩K is a Rad-supplement of V in U that is a
direct summand of U. It follows that U is a cgs⊕-module.
Let {Li}i∈I be the family of cgs⊕-submodules of M. Cgs⊕M will denote the sum
of Lis for all i ∈ I. That is Cgs⊕M =
∑
i∈I
Li. It is clear that w Loc⊕ M ⊆ Cgs⊕M.
Proposition 2.8. Let R be a ring, M be an R-module and every cgs⊕-submodule
of M be a direct summand of M. Then every maximal submodule of M has a Rad-
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 2
188 B. NISANCI, A. PANCAR
supplement that is a direct summand of M if and only if M/Cgs⊕M does not contain
a maximal submodule.
Proof. (⇒) Suppose that M/Cgs⊕M contains a maximal submodule U/Cgs⊕M.
Then U is a maximal submodule of M. By assumption, there exist V, V ′ submodules of
M such that U + V = M, U ∩ V ⊆ RadV and M = V ⊕ V ′. By [7] (Lemma 3.3) V
is w-local. Then V is a cgs⊕-module by Proposition 2.3. It follows that V ⊆ Cgs⊕M.
M/Cgs⊕M = U/Cgs⊕M, so that M = U which is a contradiction.
(⇐) Let P be a maximal submodule of M. By assumption, P does not contain
Cgs⊕M. Hence there exists a cgs⊕-module L of M such that it is not a submodule of
P is a maximal submodule of M and L * P, then M = P + L. Note that M/P ∼=
∼= L/(P ∩ L). It follows that P ∩ L is a maximal submodule of L. Then P ∩ L is a
cofinite submodule of L. By assumption, there exist X, X ′ submodules of M such that
L = (P ∩L)+X, (P ∩L)∩X ⊆ RadX and L = X⊕X ′. It follows that M = P +X
and P ∩X ⊆ RadX. Moreover by hypothesis, X is a direct summand of M. Therefore
P has a Rad-supplement that is a direct summand of M.
Theorem 2.3. Let M be an R-module such that M = M1⊕M2 is a direct sum of
submodules M1, M2. Then M2 is a cgs⊕-module if and only if there exists a submodule
K of M2 such that K is a direct summand of M, M = K + N and N ∩K ⊆ RadK
for every cofinite submodule N/M1 of M/M1.
Proof. (⇒) Let N/M1 be any cofinite submodule of M/M1. Then N is a cofinite
submodule of M and it follows that N ∩ M2 is a cofinite submodule of M2. By
hypothesis, there exist K, K ′ submodules of M2 such that M2 = (N ∩ M2) + K,
(N∩M2)∩K ⊆ RadK and M2 = K⊕K ′. Note that M = N+K and N∩K ⊆ RadK.
Since K is a direct summand of M2, then K is a direct summand of M.
(⇐) Let U be any cofinite submodule of M2. Then M2/U is finitely generated.
It follows that (U + M1)/M1 is a cofinite submodule of M/M1. By hypothesis, there
exists a submodule K of M2 such that K is a direct summand of M, M = K +U +M1
and (U + M1) ∩ K ⊆ RadK. It follows that M2 = U + K and U ∩ K ⊆ RadK.
Therefore M2 is a cgs⊕-module.
A ring R is semiperfect if R/ RadR is semisimple and idempotents can be lifted
modulo RadR. It is shown [4] (Theorem 2.9) that R is semiperfect if and only if RR is
⊕-supplemented if and only if every free R-module is ⊕-cofinitely supplemented. Now
we generalize this fact.
Theorem 2.4. Let R be any ring. Then R is semiperfect if and only if every free
R-module is a cgs⊕-module.
Proof. Let F be any free R-module. Since R is semiperfect, then RR is ⊕-cofinitely
supplemented and it follows that RR is a cgs⊕-module. By Theorem 2.1, F is a cgs⊕-
module. Conversely, suppose that every free R-module is cgs⊕. Then RR is a cgs⊕-
module. By Proposition 2.1, RR is (cofinitely) ⊕-supplemented, i.e., R is semiperfect.
Finally, we give an example of module, which is cgs⊕ but not ⊕-cofinitely supp-
lemented.
Example 2.2 (see [12], Theorem 4.3 and Remark 4.4). Let M be a biuniform mo-
dule and S = End (M). Suppose that P is the projective S-module with dim (P ) =
= (1, 0). Then P is a indecomposable w-local module. Since dim (P ) = (1, 0), P is
not finitely generated. Hence P is a cgs⊕-module but not ⊕-cofinitely supplemented.
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Received 05.05.09
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