$G$-Supplemented Modules
Following the concept of generalized small submodule, we define $g$ -supplemented modules and characterize some fundamental properties of these modules. Moreover, the generalized radical of a module is defined and the relationship between the generalized radical and the radical of a module is invest...
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| author | Koşar, B. Nebiyev, C. Sökmez, N. Косар, Б. Небієв, С. Сокмез, Н. |
| author_facet | Koşar, B. Nebiyev, C. Sökmez, N. Косар, Б. Небієв, С. Сокмез, Н. |
| author_sort | Koşar, B. |
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| collection | OJS |
| datestamp_date | 2019-12-05T09:48:59Z |
| description | Following the concept of generalized small submodule, we define $g$ -supplemented modules and characterize some fundamental properties of these modules. Moreover, the generalized radical of a module is defined and the relationship between the generalized radical and the radical of a module is investigated. Finally, the definition of amply $g$ -supplemented modules is given with some basic properties of these modules. |
| first_indexed | 2026-03-24T02:17:18Z |
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| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
B. Koşar, C. Nebiyev, N. Sökmez (Ondokuz Mayıs Univ., Turkey)
G-SUPPLEMENTED MODULES
G-ДОПОВНЕНI МОДУЛI
Following the concept of generalized small submodule, we define g-supplemented modules and characterize some
fundamental properties of these modules. Moreover, the generalized radical of a module is defined and the relationship
between the generalized radical and radical of a module is investigated. Finally, the definition of amply g-supplemented
moduleс is given with its some basic properties.
Iз застосуванням поняття узагальненого малого пiдмодуля визначено поняття g-доповнених модулiв та охарак-
теризовано деякi фундаментальнi властивостi цих модулiв. Крiм того, визначено поняття узагальненого радикала
модуля та вивчено спiввiдношення мiж узагальненим радикалом та радикалом модуля. Насамкiнець наведено
визначення поняття рясно g-доповнених модулiв та вивчено основнi властивостi цих модулiв.
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 a submodule N of M 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. A
submodule N of an R-module M is called an essential submodule and denoted by N E M in case
K ∩N 6= 0 for every submodule K 6= 0. Let M be an R-module and K be a submodule of M. K is
called a generalized small submodule of M if for every essential submodule T of M with the property
M = K + T implies that T = M, then we write K <<g M. It is clear that every small submodule
is a generalized small submodule but the converse is not true generally. Let M be an R-module.
M is called a (generalized) hollow module if every proper submodule of M is (generalized) small
in M. Here it is clear that every hollow module is generalized hollow module. The converse of
this statement is not always true. 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 ∩V << V, then V is called a supplement of U in M. M is called a supplemented module if every
submodule of M has a supplement.
Now we will give some important properties of generalized small submodules.
Lemma 1 [6]. Let M be an R-module and K, N ≤ M. Consider the following conditions:
(1) If K ≤ N and N is generalized small submodule of M, then K is a generalized small
submodule of M.
c© B. KOŞAR, C. NEBİYEV, N. SÖKMEZ, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 6 861
862 B. KOŞAR, C. NEBİYEV, N. SÖKMEZ
(2) If K is contained in N and a generalized small submodule of N, then K is a generalized
small submodule in submodules of M which contains submodule N.
(3) Let f : M → N be an R-module homomorphism. If K <<g M, then f (K) <<g M.
(4) If K <<g L and N <<g T, then K +N <<g L+ T.
Corollary 1. Let M be an R-module and K ≤ N ≤ M. If N <<g M, then N/K <<g M/K.
Corollary 2. Let M be an R-module, K <<g M and L ≤ M. Then (K + L)/L <<g M/L.
2. G-supplemented modules.
Definition 1. Let M be an R-module and U, V ≤ M. If M = U + V and M = U + T with
T E V implies that T = V, then V is called a g-supplement of U in M. If every submodule of M
has a g-supplement in M, then M is called a g-supplemented module.
Supplemented modules are g-supplemented.
Lemma 2. Let M be an R-module, U ≤ M and V ≤ M. Then V is a g-supplement of U in M
if and only if M = U + V and U ∩ V <<g V.
Proof. ( ⇒) Let U ∩ V + T = V and T E V. Then M = U + V = U + U ∩ V + T = U + T
and since V is a g-supplement of U in M and T E V, T = V. Hence U ∩ V <<g V.
(⇐) Let M = U + V and U ∩ V <<g V. Let M = U + T with T E V. Since M = U + T
and T ≤ V, by Modular Law V = V ∩M = V ∩ (U + T ) = U ∩ V + T. Then by U ∩ V <<g V,
T = V. Hence V is a g-supplement of U in M.
Lemma 3. Let M be an R-module, M1 ≤ M, U ≤ M and M1 be a g-supplemented module. If
M1 + U has a g-supplement in M, then so does U.
Proof. Let X be a g-supplement of M1 + U in M. Then M1 + U + X = M and (M1 +
+ U) ∩ X <<g X. Since M1 is g-supplemented, (U +X) ∩ M1 has a g-supplement Y in M1,
i.e., M1 ∩ (U +X) + Y = M1 and M1 ∩ (U +X) ∩ Y <<g Y. Following this, we have M =
= M1 ∩ (U +X)+Y +U +X = U +X+Y and U ∩ (X + Y ) ≤ X ∩ (U + Y )+Y ∩ (U +X) ≤
≤ X ∩ (M1 + U) + Y ∩ M1 ∩ (U +X) <<g X + Y. Hence X + Y is a g-supplement of U
in M.
Theorem 1. Let M = M1 + M2. If M1 and M2 are g-supplemented modules, then M is a
g-supplemented module.
Proof. Clear from Lemma 3.
Corollary 3. Any finite sum of g-supplemented modules are g-supplemented.
Lemma 4. Let M be an R-module, X ≤ U ≤ M and V be a g-supplement of U. Then
(V +X) /X is a g-supplement of U/X in M/X.
Proof. Since V is a g-supplement of U in M, we have M = U + V and U ∩ V <<g V.
Thus (U ∩ V +X) /X <<g (V +X) /X by Lemma 1. Since M = U + V, it is easy to see that
M/X = (U + V ) /X = U/X + (V +X) /X and U/X ∩ (V +X) /X = (U ∩ V +X) /X <<g
<<g (V +X) /X. Therefore (V +X) /X is a g-supplement of U/X in M/X.
Theorem 2. If M is a g-supplemented module, then every factor module of M is g-supplemented.
Proof. Clear from Lemma 4.
Corollary 4. If M is a g-supplemented module, then the homomorphic image of M is g-
supplemented.
Theorem 3. Let M be an R-module, K be a direct summand of M and T ≤ K. Then T <<g K
if and only if T <<g M.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 6
G-SUPPLEMENTED MODULES 863
Proof. (⇒) Clear from Lemma 1.
(⇐) Let T <<g M. Assume that M = K ⊕ Y. If we consider the canonical map π : M → K,
then we get T = π (T ) <<g π(M) = K by Lemma 1.
Definition 2. Let M be an R-module and T ≤ M. If T is both maximal and essential in M,
then T is called a generalized maximal submodule of M. The intersection of all generalized maximal
submodules of M is called the generalized radical of M denoted by Radg M. If M has not a
generalized maximal submodule, then we denote Radg M = M.
Lemma 5. Let M be an R-module. If M has at least one generalized maximal submodule, then
Radg M =
∑
L<<gM
L.
Proof. Let L <<g M. If L * T with T is a generalized maximal submodule of M, then we get
L+ T = M since T is maximal. Thus T = M, which is a contradiction. Therefore L is contained
in every generalized maximal submodule of M. Hence
∑
L<<gM
L ⊆ Radg M.
Let x ∈ Radg M. Suppose that Rx is not generalized small in M and Ω =
{
T ≤ M | x /∈
/∈ T, T E M and Rx+ T = M
}
. Since Rx is not generalized small in M, we get Ω 6= ∅. It is clear
that every chain has a upper bound by inclusion in Ω. Hence Ω contains a maximal element K by
Zorn’s lemma. We can easily show that K is a generalized maximal submodule of M. Since K ∈ Ω,
we have x /∈ K. Since Radg M ⊆ K, we get x /∈ Radg M. This is a contradiction. Therefore
Rx <<g M and then Radg M ⊆
∑
L<<gM
L. So we get Radg M =
∑
L<<gM
L.
Corollary 5. If M has no generalized maximal submodule, then Radg M =
∑
L<<gM
L.
Proof. Similar to the proof of Lemma 5.
Corollary 6. Let M be an R-module. Then RadM ≤ Radg M.
Example 1. For a non-zero simple R-module M, we have RadM = 0 6= M = Radg M.
Theorem 4. LetM be an R-module with Radg M 6= M. The following conditions are equivalent:
(i) M is a generalized hollow module,
(ii) M is a local module,
(iii) M is a hollow module.
Proof. (i) ⇒ (ii) Let M be a generalized hollow module and T be any proper submodule of M.
Then T <<g M and we have T ≤ Radg M by Lemma 5. Since Radg M 6= M, M is local and so
the proof is complete.
(ii) ⇒ (iii) Clear.
(iii) ⇒ (i) Clear.
Theorem 5. If M is a finitely generated R-module and M has a proper essential submodule,
then every proper essential submodule of M is contained in a generalized maximal submodule.
Proof. Let K be any proper essential submodule of M. Since M is finitely generated, K is
contained in a maximal submodule T and T E M due to K E M.
Theorem 6. Let M be an R-module and Radg M 6= M. If every proper essential submodule of
M is contained in a generalized maximal submodule, then Radg M <<g M.
Proof. Clear.
3. Amply G-supplemented modules.
Definition 3. Let M be an R-module and U ≤ M. If, for every V ≤ M with M = U + V, U
has a g-supplement T in M such that T ≤ V, then we say that U has ample g-supplements in M. If
every submodule of M has ample g-supplements in M, then M is called an amply g-supplemented
module.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 6
864 B. KOŞAR, C. NEBİYEV, N. SÖKMEZ
Theorem 7. Let M be an R-module, U1, U2 ≤ M and M = U1 +U2. If U1 and U2 have ample
g-supplements in M, then U1 ∩ U2 has also ample g-supplements in M.
Proof. Let U1 ∩ U2 + T = M. Then we have M = U1 + U2 ∩ T = U2 + U1 ∩ T. Since U1
and U2 have ample g-supplements in M, then U1 has a g-supplement V1 with V1 ≤ U2 ∩ T and
U2 has a g-supplement V2 with V2 ≤ U1 ∩ T. Since M = U1 + V1 and V1 ≤ U2, by Modular
Law U2 = U2 ∩ (U1 + V1) = U1 ∩ U2 + V1. Similarly we have U1 = U1 ∩ U2 + V2. Then M =
= U1 + U2 = U1 ∩ U2 + V2 + U1 ∩ U2 + V1 = U1 ∩ U2 + V1 + V2 and U1 ∩ U2 ∩ (V1 + V2) =
= U1 ∩ (V1 + U2 ∩ V2) = U1 ∩ V1 +U2 ∩ V2 <<g M. Hence V1 + V2 is a g-supplement of U1 ∩U2
and since V1 + V2 ≤ T, U1 ∩ U2 has ample g-supplements in M.
Theorem 8. If M is an amply g-supplemented module, then every factor module of M is amply
g-supplemented.
Proof. Clear.
Corollary 7. If M is an amply g-supplemented module, then the homomorphic image of M is
amply g-supplemented.
Proof. Clear from Lemma 4.
Theorem 9. Let M be an R-module. If every submodule of M is g-supplemented, then M is
amply g-supplemented.
Proof. Clear.
Lemma 6. If M is a π-projective and g-supplemented module, then M is an amply g-supplemen-
ted module.
Proof. Let M = U+V and X be a g-supplement of U. Since M is π-projective and M = U+V,
there exists an R-module homomorphism f : M → M such that Im f ⊂ V and Im(1− f) ⊂ U. So,
we have M = f(M)+ (1− f)(M) = f(U)+ f(X)+U = U + f(X). Suppose that a ∈ U ∩ f(X).
Since a ∈ f(X), then there exists x ∈ X such that a = f(x). Since a = f(x) = f(x) − x + x =
= x− (1− f)(x) and (1 − f)(x) ∈ U we obtain x = a+ (1 − f)(x) and x ∈ U. Thus x ∈ U ∩X
and so f(X) ∈ f (U ∩X) . Therefore we get U ∩ f(X) ≤ f (U ∩X) <<g f(X). This means that
f(X) is a g-supplement of U in M. Moreover f(X) ⊂ V. Therefore M is amply g-supplemented.
Now the following corollary can be easily written as a consequence of Lemma 6.
Corollary 8. If M is a projective and g-supplemented module, then M is an amply supplemented
module.
1. Büyükaşik E.,Türkmen E. Strongly radical supplemented modules // Ukr. Math. J. – 2011. – 63, № 8. – P. 1140 – 1146.
2. Clark J., Lomp C., Vanaja N., Wisbauer R. Lifting modules supplements and projectivity in module theory // Front.
Math. – Basel: Birkhäuser, 2006.
3. Kasch F. Modules and rings. – Munich: Ludwing-Maximilian Univ., 1982.
4. Lomp C. Semilocal modules and rings // Communs Algebra. – 1999. – P. 1921 – 1935.
5. Sharpe D. W., Vamos P. Injective modules. – Cambridge Univ. Press, 1972.
6. Sökmez N., Koşar B., Nebiyev C. Genelleştirilmiş Küçük Alt Modüller // XXIII Ulusal Mat. Semp. – Kayseri: Erciyes
Üniv., 2010.
7. Zöschinger H. Komplementierte Moduln über Dedekindringen // J. Algebra. – 1974. – 29. – P. 42 – 56.
8. Zöschinger H. Moduln die in jeder Erweiterung ein Komplement haben // Math. Scand. – 1974. – 35. – P. 267 – 287.
9. Zöschinger H. Basis-Untermoduln und Quasi-kotorsions-Moduln über diskrete Bewertungsringen // Bayer. Akad.
Wiss. Math.-Nat. Kl. Sitzungsber. – 1977. – P. 9 – 16.
Received 02.07.13
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 6
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| last_indexed | 2026-03-24T02:17:18Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-20282019-12-05T09:48:59Z $G$-Supplemented Modules $G$ -supplemented modules Koşar, B. Nebiyev, C. Sökmez, N. Косар, Б. Небієв, С. Сокмез, Н. Following the concept of generalized small submodule, we define $g$ -supplemented modules and characterize some fundamental properties of these modules. Moreover, the generalized radical of a module is defined and the relationship between the generalized radical and the radical of a module is investigated. Finally, the definition of amply $g$ -supplemented modules is given with some basic properties of these modules. Із застосуванням поняття узагальненого малого підмодуля визначено поняття $g$-доповнених модулів та охарактеризовано дєякі фундаментальні властивості цих модулів. Крім того, визначено поняття узагальненого радикала модуля та вивчено співвідношення між узагальненим радикалом та радикалом модуля. Насамкінець наведено визначення поняття рясно $g$-доповнених модулів та вивчено основні властивості цих модулів. Institute of Mathematics, NAS of Ukraine 2015-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2028 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 6 (2015); 861–864 Український математичний журнал; Том 67 № 6 (2015); 861–864 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2028/1074 https://umj.imath.kiev.ua/index.php/umj/article/view/2028/1075 Copyright (c) 2015 Koşar B.; Nebiyev C.; Sökmez N. |
| spellingShingle | Koşar, B. Nebiyev, C. Sökmez, N. Косар, Б. Небієв, С. Сокмез, Н. $G$-Supplemented Modules |
| title | $G$-Supplemented Modules |
| title_alt | $G$ -supplemented modules |
| title_full | $G$-Supplemented Modules |
| title_fullStr | $G$-Supplemented Modules |
| title_full_unstemmed | $G$-Supplemented Modules |
| title_short | $G$-Supplemented Modules |
| title_sort | $g$-supplemented modules |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2028 |
| work_keys_str_mv | AT kosarb gsupplementedmodules AT nebiyevc gsupplementedmodules AT sokmezn gsupplementedmodules AT kosarb gsupplementedmodules AT nebíêvs gsupplementedmodules AT sokmezn gsupplementedmodules |