• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.403-414
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• 2016

Let R be a ring, and let M be a left R-module. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Any finite direct sum of Rad-supplementing modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal, reduced and the free left R-module $(_RR)^{({\mathbb{N})}$ is Rad-supplementing if and only if R is reduced and the free left R-module $(_RR)^{({\mathbb{N})}$ is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every left R-module is (ample) Rad-supplementing if and only if R/P(R) is left perfect, where P(R) is the sum of all left ideals I of R such that Rad I = I.

• Honam Mathematical Journal
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• v.41 no.3
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• pp.531-538
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• 2019

We introduce FI-${\oplus}$-supplemented modules as a proper generalization of ${\oplus}$-supplemented modules. We prove that; (1) every finite direct sum of FI-${\oplus}$-supplemented R-modules is an FI-${\oplus}$-supplemented R-module for any ring R ; (2) if every left R-module is FI-${\oplus}$-supplemented over a semilocal ring R, then R is left perfect; (3) if M is a finitely generated torsion-free uniform R-module over a commutative integrally closed domain such that every direct summand of M is FI-${\oplus}$-supplemented, then M is a direct sum of cyclic modules.

• Honam Mathematical Journal
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• v.42 no.3
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• pp.591-600
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• 2020

A module M is called ss-semilocal if every submodule U of M has a weak supplement V in M such that U∩V is semisimple. In this paper, we provide the basic properties of ss-semilocal modules. In particular, it is proved that, for a ring R, _RR is ss-semilocal if and only if every left R-module is ss-semilocal if and only if R is semilocal and Rad(R) ⊆ Soc(RR). We define projective ss-covers and prove the rings with the property that every (simple) module has a projective ss-cover are ss-semilocal.

• Journal of the Korean Mathematical Society
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• v.51 no.6
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• pp.1305-1319
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• 2014

A submodule N of a right R-module M is called coneat if for every simple right R-module S, any homomorphism $N{\rightarrow}S$ can be extended to a homomorphism $M{\rightarrow}S$. M is called coneat-flat if the kernel of any epimorphism $Y{\rightarrow}M{\rightarrow}0$ is coneat in Y. It is proven that (1) coneat submodules of any right R-module are coclosed if and only if R is right K-ring; (2) every right R-module is coneat-flat if and only if R is right V -ring; (3) coneat submodules of right injective modules are exactly the modules which have no maximal submodules if and only if R is right small ring. If R is commutative, then a module M is coneat-flat if and only if $M^+$ is m-injective. Every maximal left ideal of R is finitely generated if and only if every absolutely pure left R-module is m-injective. A commutative ring R is perfect if and only if every coneat-flat module is projective. We also study the rings over which coneat-flat and flat modules coincide.

• The Journal of Korean Institute of Electromagnetic Engineering and Science
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• v.19 no.8
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• pp.862-877
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• 2008

In this paper, two types of wideband 3-dB ring hybrids are compared and discussed to show the ring hybrid with a set of coupled-line sections better. However, the better one still has a realization problem that perfect matching can be achieved only with -3 dB coupling power. To solve the problem, a set of coupled-line sections with two shorts is synthesized using one- and two-port equivalent circuits and design equations are derived to have perfect matching, regardless of the coupling power. Based on the design equations, a modified ${\Pi}-type$ of transmission-line equivalent circuit is newly suggested. It consists of coupled-line sections with two shorts and two open stubs and can be used to reduce a transmission-line section, especially when its electrical length is greater than ${\pi}$. Therefore, the $3\;{\lambda}/4$ transmission-line section of a conventional ring hybrid can be reduced to less than ${\pi}/2$. To verify the modified ${\Pi}-type$ of transmission- line equivalent circuit, two kinds of simulations are carried out; one is fixing the electrical length of the coupled-line sections and the other fixing its coupling coefficient. The simulation results show that the bandwidths of resulting small transmission lines are strongly dependent on the coupling power. Using modified and conventional ${\Pi}-types$ of transmission-line equivalent circuits, a small ring hybrid is built and named a compact wideband coupled-line ring hybrid, due to the fact that a set of coupled-line sections is included. One of compact ring hybrids is compared with a conventional ring hybrid and the compared results demonstrate that the bandwidth of a proposed compact ring hybrid is much wider, in spite of being more than three times smaller in size. To test the compact ring hybrids, a microstrip compact ring hybrid, whose total transmission-line length is $220^{\circ}$, is fabricated and measured. The measured power divisions($S_{21}$, $S_{41}$, $S_{23}$ and $S_{43}$) are -2.78 dB, -3.34 dB, -2.8 dB and -3.2 dB, respectively at a design center frequency of 2 GHz, matching and isolation less than -20 dB in more than 20 % fractional bandwidth.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.973-990
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• 2020

For the question "when is E(_RR) a flat left R-module for any ring R?", in this paper, we deal with a class of modules partaking in the hierarchy of injective and cotorsion modules, so-called Harmanci injective modules, which turn out by the motivation of relations among the concepts of injectivity, flatness and cotorsionness. We give some characterizations and properties of this class of modules. It is shown that the class of all Harmanci injective modules is enveloping, and forms a perfect cotorsion theory with the class of modules whose character modules are Matlis injective. For the objective we pursue, we characterize when the injective envelope of a ring as a module over itself is a flat module.

• 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.50 no.2
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• pp.389-398
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• 2013

Let R be a ring and $\mathcal{Q}$ be a quiver. In this paper we give another definition of purity in the category of quiver representations. Under such definition we prove that the class of all pure injective representations of $\mathcal{Q}$ by R-modules is preenveloping. In case $\mathcal{Q}$ is a left rooted semi-co-barren quiver and R is left Noetherian, we show that every cotorsion flat representation of $\mathcal{Q}$ is pure injective. If, furthermore, R is $n$-perfect and $\mathcal{F}$ is a flat representation $\mathcal{Q}$, then the pure injective dimension of $\mathcal{F}$ is at most $n$.