• East Asian mathematical journal
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• v.14 no.2
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• pp.259-268
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• 1998

In this paper, for a commutative ring R with an identity, considering the endomorphism ring $End_R$(M) of left R-module $_RM$ which is (quasi-)injective or (quasi-)projective, some discriminations of prime endomorphism were found as follows: each epimorphism with the irreducible(or simple) kernel on a (quasi-)injective module and each monomorphism with maximal image on a (quasi-)projective module are prime. It was shown that for a field F, any given square matrix in $Mat_{n{\times}n}$(F) with maximal image and irreducible kernel is a prime matrix, furthermore, any given matrix in $Mat_{n{\times}n}$(F) for any field F can be factored into a product of prime matrices.

• Honam Mathematical Journal
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• v.39 no.2
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• pp.275-296
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• 2017

All rings considered here are commutative rings with identity and all modules considered here are unital left modules. A submodule N of an R-module M is said to be extended to M if $N=aM$ for some ideal a of R and it is said to be fully invariant if ${\varphi}(L){\subseteq}L$ for every ${\varphi}{\in}End(M)$. An R-module M is called a [resp., fully invariant] multiplication module if every [resp., fully invariant] submodule is extended to M. The class of fully invariant multiplication modules is bigger than the class of multiplication modules. We deal with prime submodules and associated prime submodules of fully invariant multiplication modules. In particular, when M is a nonzero faithful multiplication module over a Noetherian ring, we characterize the zero-divisors of M in terms of the associated prime submodules, and we show that the set Aps(M) of associated prime submodules of M determines the set $Zdv_M(M)$ of zero-dvisors of M and the support Supp(M) of M.

• Journal of the Korean Mathematical Society
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• v.61 no.3
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• pp.603-620
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• 2024

Let 𝒱, 𝒲, 𝑦, 𝒳 be four classes of left R-modules. The notion of (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein R-complexes is introduced, and it is shown that under certain mild technical assumptions on 𝒱, 𝒲, 𝑦, 𝒳, an R-complex 𝑴 is (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein if and only if the module in each degree of 𝑴 is (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein and the total Hom complexs Hom_R(𝒀, 𝑴), Hom_R(𝑴, 𝑿) are exact for any ${\mathbf{Y}}\,{\in}\,{\tilde{\mathcal{Y}}}$ and any ${\mathbf{X}}\,{\in}\,{\tilde{\mathcal{X}}}$. Many known results are recovered, and some new cases are also naturally generated.

• Journal of the Korean Mathematical Society
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• v.60 no.6
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• pp.1233-1254
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• 2023

We introduce two subclasses of abelian McCoy rings, so-called π-CN-rings and π-duo rings, and systematically study their fundamental characteristic properties accomplished with relationships among certain classical sorts of rings such as 2-primal rings, bounded rings etc. It is shown that a ring R is π-CN whenever every nilpotent element of index 2 in R is central. These rings naturally generalize the long-known class of CN-rings, introduced by Drazin [9]. It is proved that π-CN-rings are abelian, McCoy and 2-primal. We also show that, π-duo rings are strongly McCoy and abelian and also they are strongly right AB. If R is π-duo, then R[x] has property (A). If R is π-duo and it is either right weakly continuous or every prime ideal of R is maximal, then R has property (A). A π-duo ring R is left perfect if and only if R contains no infinite set of orthogonal idempotents and every left R-module has a maximal submodule. Our achieved results substantially improve many existing results.

• 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.61 no.1
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• pp.1-12
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• 2024

In this paper, we introduce the notion of Gorenstein (m, n)-flat modules as an extension of (m, n)-flat left R-modules over a ring R, where m and n are two fixed positive integers. We demonstrate that the class of all Gorenstein (m, n)-flat modules forms a Kaplansky class and establish that (𝓖𝓕_m,n(R),𝓖𝓒_m,n(R)) constitutes a hereditary perfect cotorsion pair (where 𝓖𝓕_m,n(R) denotes the class of Gorenstein (m, n)-flat modules and 𝓖𝓒_m,n(R) refers to the class of Gorenstein (m, n)-cotorsion modules) over slightly (m, n)-coherent rings.

• Bulletin of the Korean Mathematical Society
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• v.45 no.3
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• pp.457-466
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• 2008

Let R be a right near-ring. An R-group of type-5/2 which is a natural generalization of an irreducible (ring) module is introduced in near-rings. An R-group of type-5/2 is an R-group of type-2 and an R-group of type-3 is an R-group of type-5/2. Using it $J_{5/2}$, the Jacobson radical of type-5/2, is introduced in near-rings and it is observed that $J_2(R){\subseteq}J_{5/2}(R){\subseteq}J_3(R)$. It is shown that $J_{5/2}$ is an ideal-hereditary Kurosh-Amitsur radical (KA-radical) in the class of all zero-symmetric near-rings. But $J_{5/2}$ is not a KA-radical in the class of all near-rings. By introducing an R-group of type-(5/2)(0) it is shown that $J_{(5/2)(0)}$, the corresponding Jacobson radical of type-(5/2)(0), is a KA-radical in the class of all near-rings which extends the radical $J_{5/2}$ of zero-symmetric near-rings to the class of all near-rings.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.29-40
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• 2024

Let S and R be rings and _SC_R a semidualizing bimodule. We introduce the notion of G_C-FP_n-injective modules, which generalizes G_C-FP-injective modules and G_C-weak injective modules. The homological properties and the stability of G_C-FP_n-injective modules are investigated. When S is a left n-coherent ring, several nice properties and new Foxby equivalences relative to G_C-FP_n-injective modules are given.

• Journal of the Korea Convergence Society
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• v.12 no.7
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• pp.263-271
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• 2021

Optoelectronic shooting training systems are used in shooting training sites to improve the accuracy of shooting by tracking the trajectories of gun movements. However, optoelectronic-based systems have limitations in terms of cost, complexity of installation, and the risk that electronic targets may be broken. In this study, we developed and verified a shooting training system that measures postural tremors using a low-cost accelerometer. The acceleration sensor module was designed to be attached to the air cylinder of a gun. Postural tremors were evaluated based on amplitude, frequency, and spatial pattern index, which were computed using acceleration data. The postural tremor indices between the accelerometer and optoelectronic-based system were highly correlated (left-right and up-down directions: r = 0.76 and r = 0.70, respectively). We validated the developed shooting training system using an independent two-sample t-test, which identified a significant difference (p < 0.05) in the calculated postural tremor index according to the athlete's shooting score (i.e., best and worst shots).