$T$-radical and strongly $T$-radical supplemented modules
We define (strongly) t-radical supplemented modules and investigate some properties of these modules. These modules lie between strongly radical supplemented and strongly $\oplus$ -radical supplemented modules. We also study the relationship between these modules and present examples separating stro...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507804619505664 |
|---|---|
| author | Koşar, B. Nebiyev, C. Косар, Б. Небієв, С. |
| author_facet | Koşar, B. Nebiyev, C. Косар, Б. Небієв, С. |
| author_sort | Koşar, B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:31:35Z |
| description | We define (strongly) t-radical supplemented modules and investigate some properties of these modules. These modules lie between strongly radical supplemented and strongly $\oplus$ -radical supplemented modules. We also study the relationship between these modules and present examples separating strongly $t$-radical supplemented modules, supplemented modules,
and strongly $\oplus$-radical supplemented modules. |
| first_indexed | 2026-03-24T02:15:08Z |
| format | Article |
| fulltext |
UDC 512.5
B. Koşar, C. Nebiyev (Ondokuz Mayıs Univ., Turkey)
\bfitT -RADICAL AND STRONGLY \bfitT -RADICAL SUPPLEMENTED MODULES
\bfitT -RADICAL AND STRONGLY \bfitT -RADICAL SUPPLEMENTED MODULES
We define (strongly) t-radical supplemented modules and investigate some properties of these modules. These modules
lie between strongly radical supplemented and strongly \oplus -radical supplemented modules. We also study the relationship
between these modules and present examples separating strongly t-radical supplemented modules, supplemented modules,
and strongly \oplus -radical supplemented modules.
Визначено поняття (сильно) t-радикальних доповнених модулiв та вивчено деякi властивостi цих модулiв. Тскi
модулi лежать мiж сильно радикальними доповненими та сильно \oplus -радикальними доповненими модулями. Також
вивчено спiввiдношеняя мiж цими модулями та наведено приклади, що вiддiляють сильно t-радикальнi доповненi
модулi, доповненi модулi та сильно \oplus -радикальнi доповненi модулi.
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 R0 - module. We will denote a submodule N of M by N \leq M.
Let M be an R-module and N \leq 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 \ll M. Let M be an R-module and
N \leq M. If there exists a submodule K of M such that M = N + K and N \cap K = 0, then
N is called a direct summand of M and it is denoted by M = N \oplus K [14]. \mathrm{R}\mathrm{a}\mathrm{d}M indicates
the radical of M. A submodule N of M is called radical if N has no maximal submodules, i.e.,
N = \mathrm{R}\mathrm{a}\mathrm{d}N. M is called a hollow module if every 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. 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 \cap V \ll V, then V is called a
supplement [5, 9, 16] of U in M. M is called a supplemented module if every submodule of M has
a supplement in M. A module M is called amply supplemented if V contains a supplement of U in
M whenever M = U + V [14]. Clearly every amply supplemented module is supplemented. M is
called [7, 10, 11] \oplus -supplemented 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, 13] of U in M if M = U + V and U \cap V \leq \mathrm{R}\mathrm{a}\mathrm{d}V. 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 \oplus -
supplemented [6, 10, 11] module if every submodule of M has a generalized supplement that is a direct
summand in M . A submodule N of an R-module M is called cofinite if M/N is finitely generated.
Note that M is called \pi -projective if whenever M = U + V then there exists a homomorphism
f : M \rightarrow M such that f (M) \subseteq U and (1 - f) (M) \subseteq V [14].
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 \cap (K + X) has a generalized supplement Y in N, then X + Y is a
generalized supplement of K in M.
Proof. See [6], (Lemma 3.2).
c\bigcirc BERNA KOŞAR, CELIL NEBIYEV, 2016
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1192 BERNA KOŞAR, CELIL NEBIYEV
Lemma 1.2. If V is a supplement in a module M, then \mathrm{R}\mathrm{a}\mathrm{d}V = V \cap \mathrm{R}\mathrm{a}\mathrm{d}M.
Proof. See [3] (Corollary 4.2].
Lemma 1.3. Let M be a \pi -projective module and K,L be two submodules of M. If K and L
are mutual supplements in M, then K \cap L = 0 and M = K \oplus L.
Proof. See [14] (41.14(2)).
2. T-sum and T-summand.
Definition 2.1. Let M be an R-module, U and V be two submodules of M. M is called t-sum
of U and V if U and V are mutual supplements in M., i.e., M = U + V, U \cap V \ll U and
U \cap V \ll V. Having this property of M is called a t-decomposition of M, U and V are called
t-summand of M. (see also [8]).
Theorem 2.1. Let M be an R-module. M is an amply supplemented module if and only if for
every U \leq M there exists a t-decomposition M = X + Y of M such that X \leq U and U \cap Y \ll Y.
Proof. (\Rightarrow ) Let M be an amply supplemented module. Consider any submodule U of M. Since
M is amply supplemented module, then M is supplemented module. So U has a supplement Y in
M. In this case M = U + Y and U \cap Y \ll Y. Since M = U + Y and M is amply supplemented
module, Y has a supplement X in M such that X \leq U. Therefore M is t-sum of X and Y.
(\Leftarrow ) Consider any submodule U of M and let M = U + V . By hypothesis, there exists
a t-decomposition M = X + Y of M such that X \leq U \cap V and U \cap V \cap Y \ll Y. Since
M = X + Y and X \leq U \cap V \leq V, then the modular law, V = X + V \cap Y. So we have
M = U + V = U +X + V \cap Y = U + V \cap Y. Also by hypothesis, there exists a t-decomposition
M = S + T of M such that S \leq V \cap Y and V \cap Y \cap T \ll T. Since S \leq V \cap Y and M = S + T,
then by modular law, V \cap Y = S + V \cap Y \cap T. Moreover, since V \cap Y \cap T \ll T, we get
M = U + V \cap Y = U + S + V \cap Y \cap T = U + S. In here, since U \cap S \leq U \cap V \cap Y \ll Y, then
U \cap S \ll M. Since S is a supplement in M, then U \cap S \ll S. That is, U has a supplement S in M
such that S \leq V. Therefore M is amply supplemented.
Definition 2.2. Let M be an R-module and \{ Ui\} i\in I be a collection of submodules of M. If for
every i \in I, Ui and
\sum
k\in I - \{ i\}
Uk are mutual supplements in M, then M is called t-sum of the
collection \{ Ui\} i\in I . (see also [8]).
Lemma 2.1. Let M be a \pi -projective R-module and a t-sum of U and V. Then U \cap V = 0
and M = U \oplus V.
Proof. Clear from Lemma 1.3.
The following result generalizes Lemma 2.1 which is easly proved.
Corollary 2.1. Let M be an R-module and \{ Ui\} i\in I be a collection of submodules of M. If M
is \pi -projective and a t-sum of the collection \{ Ui\} i\in I , then M = \oplus i\in IUi.
Proof. We take any k \in I. Hence Uk and
\sum
i\in I - \{ k\}
Ui are mutual supplements in M. By the
Lemma 2.1, we have Uk \cap
\Bigl( \sum
i\in I - k
Ui
\Bigr)
= 0. Therefore M = \oplus i\in IUi.
Lemma 2.2. Let M be an R-module and V be a supplement of U in M. T is a supplement of
K in V with K,T \leq V if and only if T is a supplement of U +K in M. (see also [8]).
Proof. (\Rightarrow ) Let T be a supplement of K in V. Consider any submodule T1 of T with
U +K + T1 = M. Since K,T \leq V, U +K + T1 = M and V is a supplement of U in M, then
we get K + T1 = V. Since T is a supplement of K in V, then T1 = T. So, T is a supplement of
U +K in M.
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T -RADICAL AND STRONGLY T -RADICAL SUPPLEMENTED MODULES 1193
(\Leftarrow ) Let T be a supplement of U+K in M. Consider any submodule T1 of T with K+T1 = V.
We get M = U + V = U +K + T1. Since T1 \leq T and by the assumption, we can write T1 = T.
Therefore T is a supplement of K in V.
Lemma 2.3. Let M be a t-sum of U and V. If K is a supplement of S in U and L is a
supplement of T in V, then K + L is a supplement of S + T in M (see also [8]).
Proof. Since U is a supplement of V in M and K is a supplement of S in U , by Lemma 2.2, K is
a supplement of V +S in M. Hence (V +S)\cap K \ll K . Similarly, we can prove that (U+T )\cap L \ll L.
Then (S+T )\cap (K+L) \leq (S+T+K)\cap L+(S+T+L)\cap K = (U+T )\cap L+(V +S)\cap K \ll K+L,
and by M = U + V = S +K + T +L = S + T +K +L, K +L is a supplement of S + T in M.
Lemma 2.4. Let M be a t-sum of U and V, and L, T \leq V. Then V is a t-sum of L and T if
and only if M is a t-sum of U + L and T, and M is a t-sum of U + T and L (see also [8]).
Proof. (\Rightarrow ) Let V be a t-sum of L and T . Since T is a supplement of L in V and V
is a supplement of U in M, then by Lemma 2.2, T is a supplement of U + L in M. Then
(U + L) \cap T \ll T. Similarly, we can prove that (U + T ) \cap L \ll L. Then by U \cap V \ll U,
(U + L) \cap T \leq U \cap (L + T ) + L \cap (U + T ) = U \cap V + (U + T ) \cap L \ll U + L. Since
(U + L) \cap T \ll T, (U + L) \cap T \ll U + L and M = U + V = U + L+ T, then by Definition 2.1
M is a t-sum of U + L and T. Similarly, we can prove that M is a t-sum of U + T and L.
(\Rightarrow ) Clear from Lemma 2.2.
Corollary 2.2. Let M be a t-sum of U1, U2, . . . , Un. If Ki is a supplement of Ti in Ui, i =
= 1, 2, . . . , n, then K1 +K2 + . . .+Kn is a supplement of T1 + T2 + . . .+ Tn in M (see also [8]).
Proof. Clear from Lemma 2.7.
Corollary 2.3. Let M be a t-sum of U1, U2, . . . , Un. If Ui is a t-sum of Ki and Ti, i =
= 1, 2, . . . , n, then M is a t-sum of K1 +K2 + . . .+Kn and T1 + T2 + . . .+ Tn (see also [8]).
Proof. Clear from Corollary 2.2.
Corollary 2.4. Let M be a t-sum of U1, U2, . . . , Un. If Ki is a supplement in Ui, i = 1, 2, . . . , n,
then K1 +K2 + . . .+Kn is a supplement in M (see also [8]).
Proof. Clear from Corollary 2.9.
Corollary 2.5. Let M be a t-sum of U1, U2, . . . , Un. If Ki is a t-summand of Ui, i = 1, 2, . . . , n,
then K1 +K2 + . . .+Kn is a t-summand of M (see also [8]).
Proof. Clear from Lemma 2.4.
Let M be an R-module. We say that M is called cofinitely t-generalized supplemented module
if every cofinite submodule of M has a generalized supplement such that it is a supplement in M.
Theorem 2.2. Let M be a t-sum of collection of \{ Ui\} i\in I . If for every i \in I, Ui is cofinitely
t-generalized supplemented, then M is also cofinitely t-generalized supplemented.
Proof. Let K be any cofinite submodule of M. Since M =
\sum
i\in I
Ui, then there exist
i1, i2, . . . , in \in I such that M = K + Ui1 + Ui2 + . . . + Uin . By Lemma 1.1, clearly, K has
a generalized supplement Vi1 + Vi2 + . . . + Vin in M such that Vit is a supplement in Uit for
1 \leq t \leq n. By Corollary 2.4, we get Vi1 + Vi2 + . . .+ Vin is a supplement in M. Therefore M is a
cofinitely t-generalized supplemented.
Lemma 2.5. Let M be a t-sum of collection of \{ Ui\} i\in I . Then \mathrm{R}\mathrm{a}\mathrm{d}M =
\sum
i\in I
\mathrm{R}\mathrm{a}\mathrm{d}Ui (see
also [8]).
Proof. Clearly
\sum
i\in I
\mathrm{R}\mathrm{a}\mathrm{d}Ui \leq \mathrm{R}\mathrm{a}\mathrm{d}M. Let x \in \mathrm{R}\mathrm{a}\mathrm{d}M. Since x \in M =
\sum
i\in I
\mathrm{R}\mathrm{a}\mathrm{d}Ui,
there exist i1, i2, . . . , in \in I and xit \in Uit , t = 1, 2, . . . , n such that x = xi1 + xi2 + . . . + xin .
Suppose that some submodule S of Uit for 1 \leq t \leq n with Rxit + S = Uit . In here, we can show
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1194 BERNA KOŞAR, CELIL NEBIYEV
that Rxit +S+
\sum
i\in I - \{ it\}
Ui = M. Since Rx \ll M, we have S+
\sum
i\in I - \{ it\}
Ui = M. Moreover,
since S \leq Uit and Uit is a supplement of
\sum
i\in I - \{ it\}
Ui in M, then we can write S = Uit . Hence
Rxit \ll Uit , then xit \in \mathrm{R}\mathrm{a}\mathrm{d}Uit . Therefore \mathrm{R}\mathrm{a}\mathrm{d}M \leq
\sum
i\in I
\mathrm{R}\mathrm{a}\mathrm{d}Ui.
3. (Strongly) \bfitT -radical supplemented modules.
Definition 3.1. Let M be an R-module. If the radical of M has a supplement such that is a
t-summand in M, then M is called t-radical supplemented module, that is, there exist K,L \leq M
such that M = \mathrm{R}\mathrm{a}\mathrm{d}M +K, \mathrm{R}\mathrm{a}\mathrm{d}M \cap K \ll K and M = K + L, K \cap L \ll K, K \cap L \ll L.
Definition 3.2. Let M be an R-module. If every submodule of M containing the radical of
M has a supplement that is a t-summand in M, then M is called strongly t-radical supplemented
module. That is, for every submodule K of M with \mathrm{R}\mathrm{a}\mathrm{d}M \subseteq K, there exists a t-summand L of M
such that M = K + L, K \cap L \ll L.
Lemma 3.1. Every supplemented module is strongly t-radical supplemented.
Proof. Let M be a supplemented module and let \mathrm{R}\mathrm{a}\mathrm{d}M \leq U \leq M. So U has a supplement
V in M. Since M is supplemented, V has a supplement V\' in M. Hence V and V\' are mutual
supplements in M. Therefore V is a t-summand of M. This means that M is strongly t-radical
supplemented.
In the last of this section, we will give an example of a strongly t-radical supplemented module
that is not supplemented.
Lemma 3.2. Every radical module is (strongly) t-radical supplemented.
Proof. Let M be a radical module. Clearly M has the trivial supplement 0 in M. Hence M is
t-radical supplemented. Since M is the unique submodule containing the radical, M is a strongly
t-radical supplemented.
By P (M) we denote the sum of all radical submodules of a module M. It is clear that, for any
module M, P (M) is the largest radical submodule.
Corollary 3.1. For every R-module M, P (M) is strongly t-radical supplemented.
Proof. Since \mathrm{R}\mathrm{a}\mathrm{d}P (M) = P (M) , the proof is complete.
Lemma 3.3. Let M be (strongly) t-radical supplemented module. Then M has a t-summand
which is radical.
Proof. By hypothesis, there exists V, V\'\leq M such that M = \mathrm{R}\mathrm{a}\mathrm{d}M + V, \mathrm{R}\mathrm{a}\mathrm{d}M \cap V \ll V,
M = V + V\', V \cap V\'\ll V and V \cap V\'\ll V\'. Now we prove that \mathrm{R}\mathrm{a}\mathrm{d}V\'= V\'. Since \mathrm{R}\mathrm{a}\mathrm{d}M \cap V =
= \mathrm{R}\mathrm{a}\mathrm{d}V, \mathrm{R}\mathrm{a}\mathrm{d}V \ll V. Note that \mathrm{R}\mathrm{a}\mathrm{d}M = \mathrm{R}\mathrm{a}\mathrm{d}V +\mathrm{R}\mathrm{a}\mathrm{d}V\'. So, M = V +\mathrm{R}\mathrm{a}\mathrm{d}V\'. Applying the
modular law, V\' = \mathrm{R}\mathrm{a}\mathrm{d}V\'+
\bigl(
V \cap V\'
\bigr)
. Since V \cap V\' \ll V\', then \mathrm{R}\mathrm{a}\mathrm{d}V\' = V\'. Therefore V\' is a
radical t-summand.
Recall that a module M is called reduced if P (M) = 0.
Lemma 3.4. Let M be a reduced module. If M is (strongly) t-radical supplemented, then
\mathrm{R}\mathrm{a}\mathrm{d}M \ll M.
Proof. Since M is (strongly) t-radical supplemented, there exists V, V\'\leq M, such that M =
= \mathrm{R}\mathrm{a}\mathrm{d}M +V, \mathrm{R}\mathrm{a}\mathrm{d}M \cap V \ll V and M = V +V\', V \cap V\'\ll V, V \cap V\'\ll V\'. Since \mathrm{R}\mathrm{a}\mathrm{d}M \cap V =
= \mathrm{R}\mathrm{a}\mathrm{d}V, \mathrm{R}\mathrm{a}\mathrm{d}V \ll V. By Lemma 3.3, we have \mathrm{R}\mathrm{a}\mathrm{d}V\'= V\'. Since M is reduced, P (M) = 0.
Hence we get M = V.
Lemma 3.5. Every module M with \mathrm{R}\mathrm{a}\mathrm{d}M \ll M is t-radical supplemented.
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T -RADICAL AND STRONGLY T -RADICAL SUPPLEMENTED MODULES 1195
Proof. Let M be a module with \mathrm{R}\mathrm{a}\mathrm{d}M \ll M. We assume that M = \mathrm{R}\mathrm{a}\mathrm{d}M + N for some
submodule N of M. Since \mathrm{R}\mathrm{a}\mathrm{d}M \ll M, then M = N.
An R-module M is called coatomic if every proper submodule of M is contained in a maximal
submodule of M. Note that \mathrm{R}\mathrm{a}\mathrm{d}M is small in M for every coatomic R-module M.
Corollary 3.2. Every coatomic module is t-radical supplemented.
The module RR is a maximal module if every nonzero ideal contains a maximal submodule. RR
is a left Bass module if every nonzero R-module has a maximal submodule; such rings are called left
Bass rings. R is left Bass ring if and only if for every nonzero R-module M, \mathrm{R}\mathrm{a}\mathrm{d}M \ll M. Now,
we obtain the following result.
Corollary 3.3. Every nonzero module over the left Bass ring is t-radical supplemented.
By combining the Lemma 3.1 and definitions we have the following lemma.
Lemma 3.6. Let M be an R-module with \mathrm{R}\mathrm{a}\mathrm{d}M \ll M. Then the following conditions are
equivalent.
(i) M is strongly t-radical supplemented,
(ii) M is strongly radical supplemented,
(iii) M is supplemented.
The factor modules of a strongly t-radical supplemented module need not be strongly t-radical
supplemented in general. A module M is called distributive if for every submodules K,L,N of M,
N + (K \cap L) = (N +K) \cap (N + L) or equivalently N \cap (K + L) = (N \cap K) + (N \cap L) . For
distributive modules we have the following fact.
Lemma 3.7. Let M be a distributive strongly t-radical supplemented module and U be a
submodule of M. Then M/U is strongly t-radical supplemented.
Proof. Let V/U be any submodule of M/U with \mathrm{R}\mathrm{a}\mathrm{d}(M/U) \subseteq V/U. From canonical epimor-
phism \pi : M \rightarrow M/U, we have (\mathrm{R}\mathrm{a}\mathrm{d}M + U)/U \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(M/U). So \mathrm{R}\mathrm{a}\mathrm{d}M \subseteq V. Since M is a
strongly t-radical supplemented module, then V has a supplement which is a t-summand in M. Hence
there exists T, T\'\leq M such that M = V +T, V \cap T \ll T and M = T+T\', T \cap T\'\ll T, T \cap T\'\ll T\'.
Since T is a supplement of V in M, then (T + U) /U is a supplement of V/U in M/U. Now we
show that (T + U) /U is a t-summand in M/U. From M = T + T\', we get M/U = (T + U)/U +
+ (T\'+U)/U. Since M is distributive, we have
\bigl[
(T +U) \cap (T\'+U)
\bigr]
/U =
\bigl(
U + (T \cap T\')
\bigr)
/U. On
the other hand,
\bigl(
U + (T \cap T\')
\bigr)
/U \ll (T + U)/U and
\bigl(
U + (T \cap T\')
\bigr)
/U \ll (T\'+ U)/U. Therefore
M/U is strongly t-radical supplemented.
Theorem 3.1. Let M be t-sum of M1 and M2. If M1 and M2 are t-radical supplemented, then
M is t-radical supplemented.
Proof. Since M1 is t-radical supplemented module, then \mathrm{R}\mathrm{a}\mathrm{d}M1has a supplement V1 which is
t-summand in M1. Since M2 is t-radical supplemented module, then \mathrm{R}\mathrm{a}\mathrm{d}M2 has a supplement V2
which is t-summand in M2. From M, is a t-sum of M1 and M2, by Lemma 2.5, we have \mathrm{R}\mathrm{a}\mathrm{d}M =
= \mathrm{R}\mathrm{a}\mathrm{d}M1 +\mathrm{R}\mathrm{a}\mathrm{d}M2. By Lemma 2.3, V1 + V2 is a supplement of \mathrm{R}\mathrm{a}\mathrm{d}M = \mathrm{R}\mathrm{a}\mathrm{d}M1 +\mathrm{R}\mathrm{a}\mathrm{d}M2 in
M. On the other hand, by Corollary 2.5 V1 + V2 is a t-summand in M.
Corollary 3.4. The finite t-sum of t-radical supplemented modules is t-radical supplemented.
Lemma 3.8. Let R be a nonlocal commutative domain and M be an injective R - module. Then
M is (strongly) t-radical supplemented module.
Proof. By our assumption, we can write \mathrm{R}\mathrm{a}\mathrm{d}M = M.
Over Dedekind domains, divisible modules coincide with injective modules as in Abelian groups.
Note that for a module M over a Dedekind domain R, M is divisible if and only if \mathrm{R}\mathrm{a}\mathrm{d}M = M,
and this holds if and only if M is injective; see for example [1] (Lemma 4.4).
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1196 BERNA KOŞAR, CELIL NEBIYEV
Corollary 3.5. Every module over nonlocal Dedekind domain is a submodule of (strongly) t-
radical supplemented module.
Now we give examples for to separate the structure of strongly t-radical supplemented, supple-
mented and strongly \oplus -radical supplemented module.
Example 3.1. Consider the \BbbZ -module \BbbQ . Since \mathrm{R}\mathrm{a}\mathrm{d}\BbbQ = \BbbQ , ıt follows that \BbbZ \BbbQ is strongly
t-radical supplemented. But it is well known that \BbbZ \BbbQ is not supplemented (see [7], Example 20.12).
Example 3.2. 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) \cap (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 \oplus Rx2 \oplus Rx3
R(ax1 - bx2)
= (Rx1+Rx2)\oplus Rx3. Here M is not \oplus -supplemented.
But F = Rx1 \oplus Rx2 \oplus Rx3 is completely \oplus -supplemented [7].
Since F is completely \oplus -supplemented, F is supplemented. Since a factor module of a sup-
plemented module is supplemented, we have M is supplemented. By Lemma 3.1 M is strongly
t-radical supplemented module. But M is not strongly \oplus -radical supplemented.
References
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Received 17.12.13,
after revision — 21.06.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
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| id | umjimathkievua-article-1914 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:08Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/8e/f4e9557d40c36efa829cf0f39143c68e.pdf |
| spelling | umjimathkievua-article-19142019-12-05T09:31:35Z $T$-radical and strongly $T$-radical supplemented modules $T$-radical and strongly $T$-radical supplemented modules Koşar, B. Nebiyev, C. Косар, Б. Небієв, С. We define (strongly) t-radical supplemented modules and investigate some properties of these modules. These modules lie between strongly radical supplemented and strongly $\oplus$ -radical supplemented modules. We also study the relationship between these modules and present examples separating strongly $t$-radical supplemented modules, supplemented modules, and strongly $\oplus$-radical supplemented modules. Визначено поняття (сильно) $t$-радикальних доповнених модулiв та вивчено деякi властивостi цих модулiв. Такi модулi лежать мiж сильно радикальними доповненими та сильно $\oplus$ -радикальними доповненими модулями. Також вивчено спiввiдношеняя мiж цими модулями та наведено приклади, що вiддiляють сильно $t$-радикальнi доповненi модулi, доповненi модулi та сильно $\oplus$-радикальнi доповненi модулi. Institute of Mathematics, NAS of Ukraine 2016-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1914 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 9 (2016); 1191-1196 Український математичний журнал; Том 68 № 9 (2016); 1191-1196 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1914/896 Copyright (c) 2016 Koşar B.; Nebiyev C. |
| spellingShingle | Koşar, B. Nebiyev, C. Косар, Б. Небієв, С. $T$-radical and strongly $T$-radical supplemented modules |
| title | $T$-radical and strongly $T$-radical supplemented modules |
| title_alt | $T$-radical and strongly $T$-radical supplemented modules |
| title_full | $T$-radical and strongly $T$-radical supplemented modules |
| title_fullStr | $T$-radical and strongly $T$-radical supplemented modules |
| title_full_unstemmed | $T$-radical and strongly $T$-radical supplemented modules |
| title_short | $T$-radical and strongly $T$-radical supplemented modules |
| title_sort | $t$-radical and strongly $t$-radical supplemented modules |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1914 |
| work_keys_str_mv | AT kosarb tradicalandstronglytradicalsupplementedmodules AT nebiyevc tradicalandstronglytradicalsupplementedmodules AT kosarb tradicalandstronglytradicalsupplementedmodules AT nebíêvs tradicalandstronglytradicalsupplementedmodules |