• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1281-1299
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• 2017

A submodule N of a module M is called ${\mathcal{S}}$-closed (in M) if M/N is nonsingular. It is well-known that the class Closed of short exact sequences determined by closed submodules is a proper class in the sense of Buchsbaum. However, the class $\mathcal{S}-Closed$ of short exact sequences determined by $\mathcal{S}$-closed submodules need not be a proper class. In the first part of the paper, we describe the smallest proper class ${\langle}\mathcal{S-Closed}{\rangle}$ containing $\mathcal{S-Closed}$ in terms of $\mathcal{S}$-closed submodules. We show that this class coincides with the proper classes projectively generated by Goldie torsion modules and coprojectively generated by nonsingular modules. Moreover, for a right nonsingular ring R, it coincides with the proper class generated by neat submodules if and only if R is a right SI-ring. In abelian groups, the elements of this class are exactly torsionsplitting. In the second part, coprojective modules of this class which we call ec-flat modules are also investigated. We prove that injective modules are ec-flat if and only if each injective hull of a Goldie torsion module is projective if and only if every Goldie torsion module embeds in a projective module. For a left Noetherian right nonsingular ring R of which the identity element is a sum of orthogonal primitive idempotents, we prove that the class ${\langle}\mathcal{S-Closed}{\rangle}$ coincides with the class of pure-exact sequences of modules if and only if R is a two-sided hereditary, two-sided $\mathcal{CS}$-ring and every singular right module is a direct sum of finitely presented modules.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.321-328
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• 1995

In this note all are commutative rings with identity and all modules are unital. Let R be a ring. An R-module M is called a multiplication module if for every submodule N of M there esists an ideal I of R such that N = IM. Clearly the ring R is a multiplication module as a module over itself. Also, it is well known that invertible and more generally profective ideals of R are multiplication R-modules (see [11, Theorem 1]).

• Kyungpook Mathematical Journal
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• v.59 no.3
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• pp.391-401
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• 2019

A module is said to be ${\pi}$-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this article, we focus on the class of modules having the ${\pi}$-extending property by looking at the singularity of quotient submodules. By doing so, we provide counterexamples, using hypersurfaces in projective spaces over complex numbers, to show that being generalized ${\pi}$-extending is not inherited by direct summands. Moreover, it is shown that the direct sums of generalized ${\pi}$-extending modules are generalized ${\pi}$-extending.

• Honam Mathematical Journal
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• v.30 no.4
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• pp.617-629
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• 2008

In this paper, the authors have found an equivalent condition of the endomorphism ring End(M) of a projective module M being von Neumann regular(Theorem 1.14) and found an equivalent condition of any associative ring R being von Neumann regular (Theorem 1.13).

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.163-165
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• 1994

Modules which satisfy the converse of Schur's lemma have been studied by many authors. In [6], R. Ware proved that a projective module P over a semiprime ring R is irreducible if and only if En $d_{R}$(P) is a division ring. Also, Y. Hirano and J.K. Park proved that a torsionless module M over a semiprime ring R is irreducible if and only if En $d_{R}$(M) is a division ring. In case R is a commutative ring, we obtain the following: An R-module M is irreducible if and only if En $d_{R}$(M) is a division ring and M is a multiplication R-module. Throughout this paper, R is commutative ring with identity and all modules are unital left R-modules. Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for each submodule N of M, there exists and ideal I of R such that N=IM. Cyclic R-modules are multiplication modules. In particular, irreducible R-modules are multiplication modules.dules.

• East Asian mathematical journal
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• v.10 no.2
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• pp.261-263
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• 1994

• Honam Mathematical Journal
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• v.14 no.1
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• pp.19-24
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• 1992

• Honam Mathematical Journal
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• v.36 no.2
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• pp.473-474
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• 2014

• Honam Mathematical Journal
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• v.45 no.1
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• pp.146-159
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• 2023

In this paper, we define ⊕_δ-co-coatomically supplemented and co-coatomically δ-semiperfect modules as a strongly notion of ⊕-co-coatomically supplemented and co-coatomically semiperfect modules with the help of Zhou's radical. We say that a module A is ⊕_δ-co-coatomically supplemented if each co-coatomic submodule of A has a δ-supplement in A which is a direct summand of A. And a module A is co-coatomically δ-semiperfect if each coatomic factor module of A has a projective δ-cover. Also we define co-coatomically amply δ-supplemented modules and we examined the basic properties of these modules. Furthermore, we give a ring characterization for our modules. In particular, a ring R is δ-semiperfect if and only if each free R-module is co-coatomically δ-semiperfect.

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.469-480
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• 2020

Let R be a commutative ring with identity and M a unital R-module. We give a new generalization of exact sequences called e-exact sequences. A sequence $0{\rightarrow}A{\longrightarrow[20]^f}B{\longrightarrow[20]^g}C{\rightarrow}0$ is said to be e-exact if f is monic, Imf ≤_e Kerg and Img ≤_e C. We modify many famous theorems including exact sequences to one includes e-exact sequences like 3 × 3 lemma, four and five lemmas. Next, we prove that for torsion-free module M, the contravariant functor Hom(-, M) is left e-exact and the covariant functor M ⊗ - is right e-exact. Finally, we define e-projective module and characterize it. We show that the direct sum of R-modules is e-projective module if and only if each summand is e-projective.