• Kyungpook Mathematical Journal
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• v.47 no.1
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• pp.21-30
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• 2007

We say a module $M_R$ a semicommutative module if for any $m{\in}M$ and any $a{\in}R$, $ma=0$ implies $mRa=0$. This paper gives various properties of reduced, Armendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and $M_R$ be a p.p.-module, then $M_R$ is a semicommutative module iff $M_R$ is an Armendariz module. For any ring R, R is semicommutative iff A(R, ${\alpha}$) is semicommutative. Let R be a reduced ring, it is shown that for number $n{\geq}4$ and $k=[n=2]$, $T^k_n(R)$ is semicommutative ring but $T^{k-1}_n(R)$ is not.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.267-278
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• 2024

In this paper, we continue to explore an idea presented in [3] and introduce a new class of matrix rings called staircase matrix rings which has applications in noncommutative ring theory. We show that these rings preserve the notions of reduced, symmetric, reversible, IFP, reflexive, abelian rings, etc.

• Communications of the Korean Mathematical Society
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• v.25 no.3
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• pp.349-364
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• 2010

P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a, $b\;{\in}\;R$. Commutative rings and reduced rings are reversible. In this paper, we extend the reversible condition of a ring as follows: Let R be a ring and $\alpha$ an endomorphism of R, we say that R is right (resp., left) $\alpha$-shifting if whenever $a{\alpha}(b)\;=\;0$ (resp., $\alpha{a)b\;=\;0$) for a, $b\;{\in}\;R$, $b{\alpha}{a)\;=\;0$ (resp., $\alpha(b)a\;=\;0$); and the ring R is called $\alpha$-shifting if it is both left and right $\alpha$-shifting. We investigate characterizations of $\alpha$-shifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism $\alpha$ of a ring R, R is right (resp., left) $\alpha$-shifting if and only if Q(R) is right (resp., left) $\bar{\alpha}$-shifting, whenever there exists the classical right quotient ring Q(R) of R.

• East Asian mathematical journal
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• v.33 no.1
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• pp.75-81
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• 2017

It is a well-known fact that a ring R is regular if and only if every left R-modules is flat. In this article we prove that a ring R is strongly regular if and only if every left R-modules is reduced if and only if every left-R modules is quasi-reduced.

• Korean Journal of Mathematics
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• v.21 no.1
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• pp.29-34
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• 2013

Power-Armendariz is a unifying concept of Armendariz and commutative. Let R be a ring and I be a proper ideal of R such that R/I is a power-Armendariz ring. Han et al. proved that if I is a reduced ring without identity then R is power-Armendariz. We find another direct proof of this result to see the concrete forms of various kinds of subsets appearing in the process.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.719-727
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• 2019

This article concerns a generalization of Armendariz rings that is done by restricting the degree to one. We shall call such rings, as to satisfy this property, partial-Armendariz. We first show that partial-Armendariz rings are between Armendariz rings and weak Armendariz rings. The basic structures of partial-Armendariz rings are investigated, and the relations between partial-Armendariz rings and near related ring properties are also studied.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1237-1246
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• 2022

In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it a mid ring. Also, we provide new characterizations for von Neumann regular and zero-dimensional rings. Moreover, some results about mp-ring are given. Finally, a characterization for mid rings is provided. Then it is shown that the class of mid rings is strictly between the class of reduced mp-rings (p.f. rings) and the class of mp-rings.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1083-1096
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• 2018

Let R be an associative ring. We define a subset $S_R$ of R as $S_R=\{a{\in}R{\mid}aRa=(0)\}$ and call it the source of semiprimeness of R. We first examine some basic properties of the subset $S_R$ in any ring R, and then define the notions such as R being a ${\mid}S_R{\mid}$-reduced ring, a ${\mid}S_R{\mid}$-domain and a ${\mid}S_R{\mid}$-division ring which are slight generalizations of their classical versions. Beside others, we for instance prove that a finite ${\mid}S_R{\mid}$-domain is necessarily unitary, and is in fact a ${\mid}S_R{\mid}$-division ring. However, we provide an example showing that a finite ${\mid}S_R{\mid}$-division ring does not need to be commutative. All possible values for characteristics of unitary ${\mid}S_R{\mid}$-reduced rings and ${\mid}S_R{\mid}$-domains are also determined.

• Tribology and Lubricants
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• v.25 no.5
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• pp.348-358
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• 2009

To satisfy the more severe emission regulation and the demand of higher fuel economy in near future, the combustion pressure and power output of engines is going to be higher. In order to get the reduction of engine emission and the higher power, it is needed the reduction of the tension and width of ring pack. The lower tension ring and the manufacturing technology of cast iron thinner width ring can bring the friction reduction between the ring and liner during engine running. Therfore, the fuel economy can be achieved. Thereafter the engine emission can be reduced. In this study, by using a developed basic computer program that predicts the inter-ring pressure, the motion of ring and the inter-ring pressure through a crevice volume model between adjacent rings, and the oil film thickness and the friction computed by lubrication theories, it is to be examined the effect of friction reduction from piston ring pack equipped with thinner width ring and lower tension ring.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.15 no.5
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• pp.3139-3145
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• 2014

This paper describes the design of a size-reduced ring hybrid coupler for microwave band using an artificial dielectric substrate(ADS). ADS structure adopts the second substrate on which has lots of the metalized via-holes. The effective capacitance and effective dielectric constant per unit length of ADS increases compared to the normal substrate due to the via-holes. This enables the physical length of microstrip transmission line to be reduced by adopting ADS instead of the normal substrate. In order to present an example of size-reduction of microwave wireless circuit by ADS, a size-reduced 3GHz ring hybrid coupler is designed, fabricated and measured in this work. The designed coupler has the smaller size from the normal one by 65% due to the ADS, while no critical degradation from ideal performances is observed. The measured power division ratio at two output ports are -3.05dB and -3.135dB, respectively. In addition, the phase differences are 3o for in-phase division and 176o for out of phase split. The measured performances are so similar to ideal ones, and prove the design of size-reduced ring hybrid coupler using ADS is successful.