• 대한수학회지
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• 제53권2호
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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.

• 호남수학학술지
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• 제41권3호
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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.

• 호남수학학술지
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• 제42권3호
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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.

• 대한수학회지
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• 제51권6호
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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.

• 한국전자파학회논문지
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• 제19권8호
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• pp.862-877
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• 2008

두 종류의 넓은 대역폭을 갖는 ring hybrids(하나는 coupled line이 포함되어 있고, 다른 하나는 left-handed transmission line을 포함한 ring hybrids)가 비교되었으며, 비교 결과로부터 coupled line을 포함한 ring hybrid가 모든 면에서 우수한 특성을 가짐을 보여줬다. 그러나, coupled line을 포함한 ring hybrid는 -3 dB coupling power를 가질 경우에 한해서만이 perfect matching이 이루어지기 때문에, perfect matching을 갖는 coupled line ring hybrid는 2차원으로 구현하기는 거의 불가능하다. 이 문제를 해결하기 위해서 coupled line을 해석했고, 그 해석 결과로부터 coupling coefficient에 관계없이 어느 경우에도 perfect matching을 이룰 수 있는 설계 식을 유도했다. 이 설계식을 이용하여, transmission line의 길이가 ${\pi}$보다 큰 경우에도 적용될 수 있는 크기를 줄이기 위한 새로운 형태의 transmission line 등가회로를 제시했다. 이 새로운 형태의 transmission line의 등가회로를 이용하면 기존의 ring hybrid의 $3\;{\lambda}/4$의 transmission line을 줄이는 데 사용할 수 있기 때문에 ring hybrid의 크기를 더욱 줄이는데 장점이 될 수 있다. 이 등가회로를 증명하기 위해서, coupling power를 고정하고 또는 transmission line의 길이를 고정하는 2가지 형태의 simulation을 하였으며, 대역폭은 coupled line의 coupling power에 직접적인 상관 관계가 있음을 보였다. 기존의 등가회로와 새로운 형태의 등가회로를 이용하여, 작고 넓은 대역폭을 가지는 ring hybrid를 제시하였다. 새로 제시된 ring hybrid를 이용하여, 기존의 ring hybrid와 비교하였다. 비교 결과로부터, 본 논문에서 제시한 ring hybrid의 전체 ring 둘레가 1/3보다 더 작음에도 불구하고, 대역폭이 훨씬 넓음을 보여줬다. 작고 넓은 대역폭을 가지는 ring hybrid를 측정했으며, 측정 결과는 -2.78 dB, -3.34 dB, -2.8 dB, -3.2 dB의 power division 특성을 보여줬으며, matching과 isolation은 20 % 이상의 대역폭에서 -20 dB보다 좋은 특성을 보여줬다.

• 대한수학회지
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• 제60권6호
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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.

• 대한수학회보
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• 제57권4호
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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.

• 대한수학회보
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• 제61권1호
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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.

• 대한수학회보
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• 제50권2호
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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 $n$.