t-Generalized Supplemented Modules
In the present paper, $t$-generalized supplemented modules are defined starting from the generalized ⨁-supplemented modules. In addition, we present examples separating the $t$-generalized supplemented modules, supplemented modules, and generalized ⨁-supplemented modules and also show the equality o...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508009337192448 |
|---|---|
| author | Koşar, B. Nebiyev, C. Косар, Б. Небієв, С. |
| author_facet | Koşar, B. Nebiyev, C. Косар, Б. Небієв, С. |
| author_sort | Koşar, B. |
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| collection | OJS |
| datestamp_date | 2019-12-05T09:50:14Z |
| description | In the present paper, $t$-generalized supplemented modules are defined starting from the generalized ⨁-supplemented modules. In addition, we present examples separating the $t$-generalized supplemented modules, supplemented modules, and generalized ⨁-supplemented modules and also show the equality of these modules for projective and finitely generated modules. Moreover, we define cofinitely $t$-generalized supplemented modules and give the characterization of these modules. Furthermore, for any ring $R$, we show that any finite direct sum of $t$-generalized supplemented $R$-modules is $t$-generalized supplemented and that any direct sum of cofinitely $t$-generalized supplemented $R$-modules is a cofinitely $t$-generalized supplemented module. |
| first_indexed | 2026-03-24T02:18:23Z |
| format | Article |
| fulltext |
UDC 512.5
B. Koşar, C. Nebiyev (Ondokuz Mayıs Univ., Turkey)
t-GENERALIZED SUPPLEMENTED MODULES
t-УЗАГАЛЬНЕНI ДОПОВНЕНI МОДУЛI
In this paper, t-generalized supplemented modules are defined by starting from the generalized ⊕-supplemented modules. In
addition, we present examples separating the t-generalized supplemented modules, supplemented modules, and generalized
⊕-supplemented modules and also show the equality of these modules for projective and finitely generated modules.
Moreover, we define cofinitely t-generalized supplemented modules and give the characterization of these modules.
Furthermore, for any ring R, we show that any finite direct sum of t-generalized supplemented R-modules is t-generalized
supplemented and an arbitrary direct sum of cofinitely t-generalized supplemented R-modules is a cofinitely t-generalized
supplemented module.
Доведено, що t-узагальненi доповненi модулi визначенi на основi узагальнених ⊕-доповнених модулiв. Крiм
того, наведено приклади, що вiдокремлюють t-узагальненi доповненi модулi, доповненi модулi та узагальненi
⊕-доповненi модулi, а також доведено рiвнiсть цих модулiв для проективних та скiнченнопороджених модулiв.
Також визначено кофiнiтно t-узагальненi доповненi модулi та наведено характеристику цих модулiв. Бiльш того,
для кожного кiльця R доведено, що будь-яка скiнченна пряма сума t-узагальнених доповнених R-модулiв є t-
узагальненою доповненою, а також будь-яка пряма сума кофiнiтно t-узагальнених доповнених R-модулiв є кофiнiтно
t-узагальненим доповненим R-модулем.
1. Introduction. Throughout this paper all rings will be associative with identity and all modules
will be unital left modules.
Let R be a ring and M be an R-module. We will denote a submodule N of M by N ≤ M
and a proper submodule K of M by K < M. Let M be an R-module and N ≤ M. If L = M for
every submodule L of M such that M = N + L, then N is called a small submodule of M and
denoted by N � M. Let M be an R-module and N ≤ M. If there exists a submodule K of M
such that M = N +K and N ∩K = 0, then N is called a direct summand of M and it is denoted
by M = N ⊕K. For any module M we have M = M ⊕ 0. RadM indicates the radical of M. An
R-module M is said to be simple if M have no proper submodules with distinct zero. Let M be an
R-module. M is called a (semi) hollow module if every (finitely generated) proper submodule of M
is small in M. M is called local module if M has a largest submodule, i.e., a proper submodule which
contains all other proper submodules. A module M is called distributive [10] if for every submodules
K, L, N of M, N+(K∩L) = (N+K)∩(N+L), or equivalently, N∩(K+L) = (N∩K)+(N∩L)
holds. Let U and V be submodules of M. If M = U + V and V is minimal with respect to this
property, or equivalently, M = U + V and U ∩ V � V, then V is called a supplement [12] of U
in M. M is called a supplemented module if every submodule of M has a supplement. M is called
⊕-supplemented [6, 8] module if every submodule of M has a supplement that is a direct summand
of M. Let M be an R-module and U, V be submodules of M. V is called a generalized supplement
[2, 11, 13] of U in M if M = U + V and U ∩ V ≤ RadV. M is called generalized supplemented
or briefly GS-module if every submodule of M has a generalized supplement and clearly that every
supplement submodule is a generalized supplement. M is called a generalized ⊕-supplemented [4,
5, 9, 10] module if every submodule of M has a generalized supplement that is a direct summand in
M. In this paper we generalize these modules. A submodule N of an R-module M is called cofinite
if M/N is finitely generated. M is called cofinitely supplemented [1] if every cofinite submodule
c© B. KOŞAR, C. NEBIYEV, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11 1491
1492 B. KOŞAR, C. NEBIYEV
of M has a supplement in M. M is called semiperfect module if every factor module of M has a
projective cover.
In the next section, we will define t-generalized supplemented modules and examine the rela-
tionship between these modules, supplemented modules and generalized ⊕-supplemented modules.
For any ring R, we will show that any finite direct sum of t-generalized supplemented modules is
a t-generalized supplemented module and find conditions for t-generalized supplemented modules
which make factor modules of these t-generalized supplemented modules.
In the last section, we will define cofinitely t-generalized supplemented modules and investigate
the relationship with cofinitely supplemented modules. We also show that any direct sum of cofinitely
t-generalized supplemented R-modules is also a cofinitely t-generalized supplemented R-module for
any ring R.
Lemma 1.1. Let M be an R-module and N, K be submodules of M. If N+K has a generalized
supplement X in M and N ∩ (K +X) has a generalized supplement Y in N, then X + Y is a
generalized supplement of K in M.
Proof. See [4], Lemma 3.2.
Lemma 1.2. Let M be a projective module. Consider the following conditions:
(i) M is a semiperfect module.
(ii) M is a generalized ⊕-supplemented module.
Then (i) ⇒ (ii) holds and if M is a finitely generated module then (ii) ⇒ (i) also holds.
Proof. See [10], Lemma 2.2.
2. t-Generalized supplemented modules.
Definition 2.1. Let M be an R-module. M is called a t-generalized supplemented module if
every submodule of M has a generalized supplement which is also a supplement in M. Clearly
generalized ⊕-supplemented modules are t-generalized supplemented. But the converse implication
fails to be true. This will be shown in Example 2.4.
It is also clear that although every supplemented module is a t-generalized supplemented the
converse of this statement is not always true. We will show this situation in Examples 2.1 – 2.3.
Since hollow and local modules are supplemented, they are t-generalized supplemented modules.
It is well-known that every ⊕-supplemented module is generalized ⊕-supplemented (see [4],
Example 3.11). Now we will give a situation when the converse is true.
Lemma 2.1. If M is a finitely generated R-module then M is generalized ⊕-supplemented if
and only if M is ⊕-supplemented.
Proof. (⇒) Let N be a submodule of M. Since M is generalized ⊕-supplemented, there
exists a generalized supplement K of N such that K is a direct summand in M. Hence there exists
submodules K and L of M such that M = N +K, N ∩K ≤ RadK and M = K ⊕L. Since M is
finitely generated, we have K is finitely generated and RadK � K. Therefore N ∩K � K and K
is a supplement of N in M. As a result M is ⊕-supplemented.
(⇐) Clear.
The following lemma will be used to prove Theorem 2.1.
Lemma 2.2. Let M be an R-module with M = M1⊕M2 and K, L be submodules of M1 such
that K is a supplement of L in M1. Then K is a supplement of M2 + L in M.
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t-GENERALIZED SUPPLEMENTED MODULES 1493
Proof. Let M2 + L+N = M with N ≤ K. Hence M1 = M1 ∩M = M1 ∩ (L+N +M2) =
= L+N +(M1 ∩M2) = L+N. Since N ≤ K and K is a supplement of L in M1, we get N = K.
Therefore K is a supplement of M2 + L in M.
Lemma 2.3. Let M = M1⊕M2. If K is a supplement submodule in M1 and T is a supplement
submodule in M2, then K + T is a supplement submodule in M.
Proof. Suppose that K is a supplement of U in M1 and T is a supplement of V in M2. In
this case M1 = U + K, U ∩ K � K and M2 = V + T, V ∩ T � T. Since M1 = U + K and
M2 = V +T, M = M1+M2 = U +V +K+T. It is easy to check that (U +K+V )∩T � T and
(V +T +U)∩K � K. Hence (U+V )∩(K+T ) ⊆ (U+V +T )∩K+(U+V +K)∩T � K+T.
Therefore K + T is a supplement of U + V in M.
The next result generalizes Lemma 2.3 which is easily proved.
Corollary 2.1. Let M = M1⊕M2⊕ . . .⊕Mn. For 1 ≤ i ≤ n, if Ki is a supplement submodule
in Mi, K1 +K2 + . . .+Kn is a supplement submodule in M.
Theorem 2.1. For any arbitrary ring R, the finite direct sum of t-generalized supplemented
R-modules is t-generalized supplemented.
Proof. Let n be any positive integer, {Mi}1≤i≤n be any finite collection of t-generalized
supplemented R-modules and M = M1 ⊕M2 ⊕ . . .⊕Mn. Assume that n = 2. Let M = M1 ⊕M2
and N be any submodule of M. Then M = M1+M2+N . Since M2 is t-generalized supplemented,
we can say that M2 ∩ (M1 +N) has a generalized supplement K in M2 such that K is a supplement
in M2. So K is a generalized supplement of M1+N in M. Since M1 is t-generalized supplemented,
M1 ∩ (K +N) has a generalized supplement L in M1 such that L is a supplement in M1. Thus we
get K+L is a generalized supplement of N in M (see [4]). Since K is a supplement in M2 and L is
a supplement in M1, then by Lemma 2.3 K +L is a supplement in M. Therefore M is t-generalized
supplemented. The rest of the proof can be completed by induction on n.
The relationship between the concepts “t-generalized supplemented” and ”supplemented” is ex-
pressed in the following lemma.
Lemma 2.4. Let M be a finitely generated module. Then M is t-generalized supplemented if
and only if M is supplemented.
Proof. (⇒) Let N be any submodule of M. Since M is t-generalized supplemented then there
exists K ≤ M such that M = N +K, N ∩K ⊆ RadK and K is a supplement in M. Since M is
finitely generated, we obtain RadM �M. Hence N ∩K ⊆ RadK ⊆ RadM �M and it follows
that N ∩K � K. This means that K is a supplement of N in M and so M is supplemented.
(⇐) Clear from definitions.
Lemma 2.5. Let M be an R-module. If RadM = M, then M is t-generalized supplemented.
Proof. Let N be any submodule of M. Since N +M = M and N ∩M ⊆ M = RadM , we
get that M is a generalized supplement of N. On the other hand M is a supplement of 0. Hence M
is a t-generalized supplemented.
It is easy to see that every semihollow module is t-generalized supplemented. Now we give
some examples of modules, which is t-generalized supplemented but not supplemented. Thus the
following examples are given to separate the structures of t-generalized supplemented, supplemented
and generalized ⊕-supplemented.
Example 2.1. Consider the Z-module Q. Since Q has no maximal submodule, we have RadQ =
= Q. By Lemma 2.5, Q is t-generalized supplemented module. But it is well known that Q is not
supplemented (see [3], Example 20.12).
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1494 B. KOŞAR, C. NEBIYEV
Example 2.2. Let M be a non-torsion Z-module with RadM = M. Since RadM = M then
M is t-generalized supplemented. But M is not supplemented [14].
Example 2.3. Consider the Z-module M = Q ⊕ Z/pZ, for any prime p. In this case RadM 6=
6= M. Moreover, M is t-generalized supplemented but not supplemented [4].
Example 2.4. Let R be a commutative local ring which is not a valuation ring. Let a and b
be elements of R, where neither of them divides the other. By taking a suitable quotient ring, we
may assume that (a) ∩ (b) = 0 and am = bm = 0 where m is the maximal ideal of R. Let F be
a free R-module with generators x1, x2, and x3, K be the submodule generated by ax1 − bx2 and
M = F/K. Thus,
M =
Rx1 ⊕Rx2 ⊕Rx3
R (ax1 − bx2)
= (Rx1 +Rx2)⊕Rx3.
Here M is not ⊕-supplemented. But F = Rx1 ⊕Rx2 ⊕Rx3 is completely ⊕-supplemented [6].
Since F is completely ⊕-supplemented, F is supplemented. Since a factor module of a sup-
plemented module is supplemented, we have M is supplemented. So M is t-generalized supple-
mented. Separately, since M is finitely generated and not ⊕-supplemented, M is not generalized
⊕-supplemented by Lemma 2.1.
Lemma 2.6. Let M = M1⊕M2. Then M2 is t-generalized supplemented if and only if for every
submodule N/M1 of M/M1, there exists a supplement K in M such that K ≤ M2, M = K + N
and N ∩K ⊆ RadM.
Proof. (⇒) Assume that M2 is t-generalized supplemented. Let N/M1 ≤ M/M1. Since
M2 is t-generalized supplemented, there exists a generalized supplement module K of N ∩ M2
such that K is a supplement in M2. Hence there exists K ′ ≤ M2 such that M2 = N ∩M2 + K,
N ∩M2 ∩K ⊆ RadK and M2 = K +K ′, K ∩K ′ � K. The equality M = M1 +M2 implies that
M = M1 +N ∩M2 +K = N +K. On the other hand N ∩K ⊆ RadM. Since K is a supplement
of K ′ in M2 and M = M1⊕M2, we obtain that K is a supplement of M1+K ′ in M by Lemma 2.2.
Therefore K is a supplement in M.
(⇐) Suppose that M/M1 satisfies hypothesis properties. Let H ≤M2. Consider the submodule
(H ⊕M1) /M1 ≤ M/M1. By hypothesis, there exists a supplement L in M such that L ≤ M2,
M = (L+H)⊕M1 and L ∩ (H +M1) ⊆ RadM. Since L ∩H ≤ L ∩ (H +M1) ⊆ RadM and
L ∩H ≤ L, we have L ∩H ≤ L ∩ RadM = RadL. Hence L is a generalized supplement of H
in M2.
Suppose that L is a supplement of T in M. In case M = T + L and T ∩ L � L. Note that
M2 = M2 ∩M = M2 ∩ (L+ T ) = L+M2 ∩ T. Since M2 ∩ T ∩ L ≤ T ∩ L� L, it is easy to see
that L is a supplement of M2 ∩ T in M2.
The following theorem can be written as a consequence of Lemma 2.6.
Theorem 2.2. Let M = M1 ⊕M2 be a t-generalized supplemented module and K ∩M2 be
a supplement in M for every supplement K in M with M = K + M2. Then M2 is t-generalized
supplemented.
Proof. Assume that N/M1 ≤ M/M1. Consider the submodule N ∩M2 of M. Since M is
t-generalized supplemented, there exists a generalized supplement K ′ of N ∩M2 such that K ′ is
a supplement in M, i.e., there exists a supplement K ′ in M such that M = (N ∩M2) + K ′ and
(N ∩M2)∩K ′ ≤ RadK ′. Since M = (N ∩M2) +K ′, we get M = M2 +K ′. Let K = M2 ∩K ′.
Then M2 = (N ∩M2) + (M2 ∩K ′) = (N ∩M2) + K. From M = M1 + M2 and M1 ≤ N, we
have M = N +M2 = N + (N ∩M2) + (M2 ∩K ′) = N +K. Since M = M2 +K ′ and K ′ is a
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
t-GENERALIZED SUPPLEMENTED MODULES 1495
supplement in M, K = K ′∩M2 is a supplement in M by hypothesis. Therefore M2 is t-generalized
supplemented by Lemma 2.6.
Now we will investigate some conditions which will ensure that a factor module of a (distributive)
t-generalized supplemented module is t-generalized supplemented.
Lemma 2.7. Let M be a t-generalized supplemented module and N ≤ M. If (N +K) /N
is a supplement submodule in M/N for every supplement submodule K in M, then M/N is a
t-generalized supplemented.
Proof. For any submodule X of M containing N, since M is t-generalized supplemented, there
exists D′ ≤ M such that M = X + D = D + D′, X ∩ D ≤ RadD and D ∩ D′ � D for some
submodule D of M. Since M = X+D and N ≤ X, M/N = (X +D) /N = X/N +(D +N) /N.
Note that X ∩ D ≤ RadD, X/N ∩ (D + N)/N = (X ∩D +N) /N ≤ (RadD +N) /N ≤
≤ Rad((D +N)/N). This implies that (D +N)/N is a generalized supplement of X/N in M/N.
On the other hand, D is a supplement in M and (D+N)/N is a supplement in M/N by hypothesis.
Therefore (D + N)/N is a generalized supplement of X/N in M/N such that (D + N)/N is a
supplement in M/N. Hence M/N is t-generalized supplemented.
Theorem 2.3. Let M be a distributive t-generalized supplemented module. Then for every
submodule N of M, M/N is t-generalized supplemented.
Proof. Let D be a supplement submodule in M. Then there exists D′ ≤M such that M = D+D′
and D ∩ D′ � D. Since M = D + D′, we can write that M/N = (D + N)/N + (D′ + N)/N.
From M is distributive, N + (D ∩ D′) = (N + D) ∩ (N + D′). This implies that (D + N)/N ∩
∩ (D′ + N)/N = [(D +N) ∩ (D′ +N)] /N = (N + (D ∩D′)) /N. Note that D ∩D′ � D. So,
we get (D+N)/N ∩ (D′+N)/N = (D∩D′+N)/N � (D+N)/N. Hence for every supplement
submodule D in M, (D + N)/N is a supplement submodule in M/N. Therefore by Lemma 2.7
M/N is t-generalized supplemented.
3. Cofinitely t-generalized supplemented modules.
Definition 3.1. Let M be an R-module. We say that M is called cofinitely t-generalized sup-
plemented module if every cofinite submodule of M has a generalized supplement such that it is a
supplement in M.
Clearly every cofinitely generalized ⊕-supplemented modules are cofinitely t-generalized supple-
mented.
Lemma 3.1. Let M be a finitely generated module. Then M is t-generalized supplemented if
and only if M is cofinitely t-generalized supplemented.
Proof. Since M is finitely generated, the proof is clear.
Lemma 3.2. Let M be an R-module and RadM � M. Then M is cofinitely t-generalized
supplemented if and only if M is cofinitely supplemented.
Proof. (⇒) Let N be any cofinite submodule of M. Since M is cofinitely t-generalized
supplemented, there exists K ≤M such that M = N +K, N ∩K ⊆ RadK and K is a supplement
in M. Since N∩K ⊆ RadK ⊆ RadM and RadM �M, N∩K �M. Hence we get N∩K � K.
So K is a supplement of N in M. Therefore M is cofinitely supplemented.
(⇐) Since M is cofinitely supplemented, for any cofinite submodule N of M, there exists K ≤M
such that M = N +K and N ∩K � K. From N ∩K ⊆ RadK, K is a generalized supplement of
N in M. Therefore M is cofinitely t-generalized supplemented.
As a result of Lemma 3.2, we can obtain the following corollary.
Corollary 3.1. Let M be a finitely generated R-module. M is cofinitely t-generalized supple-
mented if and only if M is cofinitely supplemented.
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1496 B. KOŞAR, C. NEBIYEV
The Corollary 2.1 together with Lemma 2.2 gives the following important theorem.
Theorem 3.1. For any ring R, the arbitrary direct sum of cofinitely t-generalized supplemented
R-modules is cofinitely t-generalized supplemented.
Proof. Let {Mi}i∈I be any collection of cofinitely t-generalized supplemented R-modules and
M = ⊕
i∈I
Mi. Let N be any cofinite submodule of M. In this case M/N is finitely generated and
there exists k ∈ Z+, xi ∈ M, 1 ≤ i ≤ k, such that M/N =
〈
{x1 +N, x2 +N, . . . , xk +N}
〉
. So
M = Rx1+Rx2+. . .+Rxk+N. In here, there exists finitely subset F = {i1, i2, . . . , in} of I such that
xi ∈ ⊕
j∈F
Mj for every 1 ≤ i ≤ k. Hence it is clear that M = Mi1 +
(
N +
∑n
j=2
Mij
)
has trivially
a generalized supplement 0 in M. Consider the submodule Mi1 ∩
(
N +
∑n
j=2
Mij
)
≤ Mi1 . Since
Mi1/
[
Mi1 ∩
(
N +
∑n
j=2
Mij
)]
∼= M/
(
N +
∑n
j=2
Mij
)
∼= (M/N) /
((
N +
∑n
j=2
Mij
)
/N
)
,
Mi1 ∩
(
N +
∑n
j=2
Mij
)
is a cofinite submodule of Mi1 . From Mi1 is cofinitely t-generalized
supplemented then Mi1 ∩
(
N +
∑n
j=2
Mij
)
has a generalized supplement Si1 such that Si1 is a
supplement in Mi1 . By Lemma 1.1, Si1 is a generalized supplement of N +
∑n
j=2
in M. Similarly
we can show that for 1 ≤ j ≤ n, N has a generalized supplement Si1 + Si2 + . . . + Sin such that
Sij is a supplement in Mij . In this case by Corollary 2.1, Si1 + Si2 + . . . + Sin is a supplement
in Mi1 ⊕ Mi2 ⊕ . . . ⊕ Min . Since Mi1 ⊕ Mi2 ⊕ . . . ⊕ Min is direct summand of M, then by
Lemma 2.2, Si1+Si2+. . .+Sin is a supplement in M. Consequently, ⊕
i∈I
Mi is cofinitely t-generalized
supplemented.
Theorem 3.2. Let M be a projective and finitely generated R-module. Then the following
assertions are equivalent:
(i) M is a semiperfect module,
(ii) M is generalized ⊕-supplemented,
(iii) M is cofinitely generalized ⊕-supplemented,
(iv) M is t-generalized supplemented,
(v) M is cofinitely t-generalized supplemented.
Proof. (i)⇔ (ii) Clear by Lemma 1.2.
(ii) ⇒ (iv) Clear from definitions.
(iv) ⇒ (ii) Since M is t-generalized supplemented and finitely generated, by Lemma 2.4 M is
supplemented. On the other hand, since M is projective then M is ⊕-supplemented. Hence M is
generalized ⊕-supplemented.
Since M is finitely generated then (ii)⇔ (iii) and (iv) ⇔ (v) are clear.
The following remark shows that a cofinitely t-generalized supplemented module need not to be
cofinitely generalized ⊕-supplemented.
Remark 3.1. In Example 2.2, fromM is t-generalized supplemented, M is cofinitely t-generalized
supplemented. But since M is finitely generated and not generalized ⊕-supplemented, M is not
cofinitely generalized ⊕-supplemented.
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Received 03.07.13
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
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| id | umjimathkievua-article-2084 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:18:23Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9c/3a3a9f7caf9731e1e633a562dc0ea49c.pdf |
| spelling | umjimathkievua-article-20842019-12-05T09:50:14Z t-Generalized Supplemented Modules t-узагальнені доповнені модулі Koşar, B. Nebiyev, C. Косар, Б. Небієв, С. In the present paper, $t$-generalized supplemented modules are defined starting from the generalized ⨁-supplemented modules. In addition, we present examples separating the $t$-generalized supplemented modules, supplemented modules, and generalized ⨁-supplemented modules and also show the equality of these modules for projective and finitely generated modules. Moreover, we define cofinitely $t$-generalized supplemented modules and give the characterization of these modules. Furthermore, for any ring $R$, we show that any finite direct sum of $t$-generalized supplemented $R$-modules is $t$-generalized supplemented and that any direct sum of cofinitely $t$-generalized supplemented $R$-modules is a cofinitely $t$-generalized supplemented module. Доведено, що $t$-узагальнені доповнені модулі визначені на основі узагальнених ⨁-доповнених модулів. Kpiм того, наведено приклади, що відокремлюють $t$-узагальнені доповнені модулі, доповнені модулі та узагальнені ⨁-доповнені модулі, а також доведено рівність цих модулів для проективних та скінченнопороджених модулів. Також визначено кофінітно $t$-узагальнені доповнені модулі та наведено характеристику цих модулів. Більш того, для кожного кільця R доведено, що будь-яка скінченна пряма сума $t$-узагальнених доповнених $R$-модулів є $t$-узагальненою доповненою, а також будь-яка пряма сума кофінітно $t$-узагальнених доповнених $R$-модулів є кофінітно $t$-узагальненим доповненим $R$-модулем. Institute of Mathematics, NAS of Ukraine 2015-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2084 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 11 (2015); 1491-1497 Український математичний журнал; Том 67 № 11 (2015); 1491-1497 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2084/1183 https://umj.imath.kiev.ua/index.php/umj/article/view/2084/1184 Copyright (c) 2015 Koşar B.; Nebiyev C. |
| spellingShingle | Koşar, B. Nebiyev, C. Косар, Б. Небієв, С. t-Generalized Supplemented Modules |
| title | t-Generalized Supplemented Modules |
| title_alt | t-узагальнені доповнені модулі |
| title_full | t-Generalized Supplemented Modules |
| title_fullStr | t-Generalized Supplemented Modules |
| title_full_unstemmed | t-Generalized Supplemented Modules |
| title_short | t-Generalized Supplemented Modules |
| title_sort | t-generalized supplemented modules |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2084 |
| work_keys_str_mv | AT kosarb tgeneralizedsupplementedmodules AT nebiyevc tgeneralizedsupplementedmodules AT kosarb tgeneralizedsupplementedmodules AT nebíêvs tgeneralizedsupplementedmodules AT kosarb tuzagalʹnenídopovnenímodulí AT nebiyevc tuzagalʹnenídopovnenímodulí AT kosarb tuzagalʹnenídopovnenímodulí AT nebíêvs tuzagalʹnenídopovnenímodulí |