• Honam Mathematical Journal
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• v.20 no.1
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• pp.15-19
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• 1998

In this note we characterize the radical of a submodule by its invelope under some conditions and prove propositions about closed submodules.

• Honam Mathematical Journal
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• v.42 no.3
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• pp.511-520
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• 2020

A module M is called cofinitely closed weak δ-supplemented (briefly δ-ccws-module) if for any cofinite closed submodule N of M has a weak δ-supplement in M. In this paper we investigate the basic properties of δ-ccws modules. In the light of this study, we can list the main facts obtained as following: (1) Any cofinite closed direct summand of a δ-ccws module is also a δ-ccws module; (2) Let R be a left δ-V -ring. Then R is a δ-ccws module iff R is a ccws-module iff R is extending; (3) Any nonsingular homomorphic image of a δ-ccws-module is also a δ-ccws-module; (4) We characterize nonsingular δ-V -rings in which all nonsingular modules are δ-ccws.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.1053-1066
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• 2010

In this paper, several properties of endomorphism rings of modules are investigated. A multiplication module M over a commutative ring R induces a commutative ring $M^*$ of endomorphisms of M and hence the relation between the prime (maximal) submodules of M and the prime (maximal) ideals of $M^*$ can be found. In particular, two classes of ideals of $M^*$ are discussed in this paper: one is of the form $G_{M^*}\;(M,\;N)\;=\;\{f\;{\in}\;M^*\;|\;f(M)\;{\subseteq}\;N\}$ and the other is of the form $G_{M^*}\;(N,\;0)\;=\;\{f\;{\in}\;M^*\;|\;f(N)\;=\;0\}$ for a submodule N of M.

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.993-999
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• 2023

Let A be a G-graded commutative ring with identity and M a graded A-module. Let m, n be positive integers with m > n. A proper graded submodule L of M is said to be graded (m, n)-closed if a^m_g·x_t ∈ L implies that aⁿ_g·x_t ∈ L, where a_g ∈ h(A) and x_t ∈ h(M). The aim of this paper is to explore some basic properties of these class of submodules which are a generalization of graded (m, n)-closed ideals. Also, we investigate GC^m_n - rad property for graded submodules.

• Kyungpook Mathematical Journal
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• v.50 no.1
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• pp.101-107
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• 2010

A module M is said to satisfy the $EC_{11}$ condition if every ec-submodule of M has a complement which is a direct summand. We show that for a multiplication module over a commutative ring the $EC_{11}$ and P-extending conditions are equivalent. It is shown that the $EC_{11}$ property is not inherited by direct summands. Moreover, we prove that if M is an $EC_{11}$-module where SocM is an ec-submodule, then it is a direct sum of a module with essential socle and a module with zero socle. An example is given to show that the reverse of the last result does not hold.

• Communications of the Korean Mathematical Society
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• v.14 no.1
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• pp.1-12
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• 1999

The faithful bi-module \ulcornerM\ulcorner with its endomorphism ring End\ulcorner(M) such that M\ulcorner is flat (in other words, End\ulcorner(M)-flat, or endo-flat)and with a commutative ring R containing an identity has been studied in this paper. The chain conditions of a faithful endo-flat module \ulcornerM relative to those of the endomorphism ring End\ulcorner(M) having the zero annihilator of each non-zero endomorphism are studied.

• Kyungpook Mathematical Journal
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• v.60 no.1
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• pp.45-51
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• 2020

A module M is called FI-extending if every fully invariant submodule of M is essential in a direct summand of M. In this work, we define a module M to be generalized FI-extending (GFI-extending) if for any fully invariant submodule N of M, there exists a direct summand D of M such that N ≤ D and that D/N is singular. The classes of FI-extending modules and singular modules are properly contained in the class of GFI-extending modules. We first develop basic properties of this newly defined class of modules in the general module setting. Then, the GFI-extending property is shown to carry over to matrix rings. Finally, we show that the class of GFI-extending modules is closed under direct sums but not under direct summands. However, it is proved that direct summands are GFI-extending under certain restrictions.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1387-1408
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• 2022

Let R be a commutative ring with a non-zero identity, S be a multiplicatively closed subset of R and M be a unital R-module. In this paper, we define a submodule N of M with (N :_R M)∩S = ∅ to be weakly S-prime if there exists s ∈ S such that whenever a ∈ R and m ∈ M with 0 ≠ am ∈ N, then either sa ∈ (N :_R M) or sm ∈ N. Many properties, examples and characterizations of weakly S-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly S-prime.

• Communications of the Korean Mathematical Society
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• v.10 no.4
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• pp.811-813
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• 1995

The purpose of this paper is to prove the divisibility of a direct injective module and every closed submodule of a $\pi$-injective module M is a direct summand of M.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.453-468
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• 2022

In this article, we define a module M to be G^z-extending if and only if for each z-closed submodule X of M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We investigate structural properties of G^z-extending modules and locate the implications between the other extending properties. We deal with decomposition theory as well as ring and module extensions for G^z-extending modules. We obtain that if a ring is right G^z-extending, then so is its essential overring. Also it is shown that the G^z-extending property is inherited by its rational hull. Furthermore it is provided some applications including matrix rings over a right G^z-extending ring.