• 대한수학회보
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• 제57권6호
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• pp.1511-1528
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• 2020

A skew generalized power series ring R[[S, 𝜔]] consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action 𝜔 of the monoid S on the ring R. Special cases of the skew generalized power series ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal'cev-Neumann series rings, the "untwisted" versions of all of these, and generalized power series rings. In this paper we obtain some necessary conditions on R, S and 𝜔 such that the skew generalized power series ring R[[S, 𝜔]] is (uniquely) clean. As particular cases of our general results we obtain new theorems on skew Mal'cev-Neumann series rings, skew Laurent series rings, and generalized power series rings.

• 대한수학회보
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• 제60권6호
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• pp.1463-1475
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• 2023

In this paper, we introduce a class of rings generalizing unit regular rings and being a subclass of semipotent rings, which is called idempotent unit regular. We call a ring R an idempotent unit regular ring if for all r ∈ R - J(R), there exist a non-zero idempotent e and a unit element u in R such that er = eu, where this condition is left and right symmetric. Thus, we have also that there exist a non-zero idempotent e and a unit u such that ere = eue for all r ∈ R - J(R). Various basic characterizations and properties of this class of rings are proved and it is given the relationships between this class of rings and some well-known classes of rings such as semiperfect, clean, exchange and semipotent. Moreover, we obtain some results about when the endomorphism ring of a module in a class of left R-modules X is idempotent unit regular.

• 대한수학회보
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• 제51권3호
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• pp.813-822
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• 2014

A ring R is called quasipolar if for every a 2 R there exists $p^2=p{\in}R$ such that $p{\in}comm^2{_R}(a)$, $ a+p{\in}U(R)$ and $ap{\in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${\pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{\times}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{\times}2$ matrix ring to be quasipolar are obtained.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제25권2호
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• pp.115-125
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• 2018

In this paper, we introduce the notion of a clean ordered Krasner hyperring and investigate some properties of it. Now, let (R, +, ${\cdot}$, ${\leq}$) be a clean ordered Krasner hyperring. The following is a natural question to ask: Is there a strongly regular relation ${\sigma}$ on R for which $R/{\sigma}$ is a clean ordered ring? Our motivation to write the present paper is reply to the above question.

• 대한수학회보
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• 제55권3호
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• pp.913-920
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• 2018

Let R be a 2-primal strongly 2-nil-clean ring. We prove that every square matrix over R is the sum of a tripotent and a nilpotent matrices. The similar result for rings of bounded index is proved. We thereby provide a large class of rings over which every matrix is the sum of a tripotent and a nilpotent matrices.

• 대한수학회보
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• 제56권1호
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• pp.57-71
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• 2019

Let R be an associative ring with identity, M a t.u.p. monoid with only one unit and ${\omega}:M{\rightarrow}End(R)$ a monoid homomorphism. Let R be a reversible, M-compatible ring and ${\alpha}=a_1g_1+{\cdots}+a_ng_n$ a non-zero element in skew monoid ring $R{\ast}M$. It is proved that if there exists a non-zero element ${\beta}=b_1h_1+{\cdots}+b_mh_m$ in $R{\ast}M$ with ${\alpha}{\beta}=c$ is a constant, then there exist $1{\leq}i_0{\leq}n$, $1{\leq}j_0{\leq}m$ such that $g_{i_0}=e=h_{j_0}$ and $a_{i_0}b_{j_0}=c$ and there exist elements a, $0{\neq}r$ in R with ${\alpha}r=ca$. As a consequence, it is proved that ${\alpha}{\in}R*M$ is unit if and only if there exists $1{\leq}i_0{\leq}n$ such that $g_{i_0}=e$, $a_{i_0}$ is unit and aj is nilpotent for each $j{\neq}i_0$, where R is a reversible or right duo ring. Furthermore, we determine the relation between clean and nil clean elements of R and those elements in skew monoid ring $R{\ast}M$, where R is a reversible or right duo ring.

• Kyungpook Mathematical Journal
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• 제63권4호
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• pp.561-569
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• 2023

An element a ∈ R is strongly rad-clean provided that there exists an idempotent e ∈ R such that a - e ∈ U(R), ae = ea and eae ∈ J(eRe). In this article, we completely determine when a 2 × 2 matrix over a commutative local ring is strongly rad clean. An application to matrices over power-series is also given.

• Kyungpook Mathematical Journal
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• 제45권4호
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• pp.519-526
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• 2005

In this paper, we prove that every exchange ring can be characterized by potent elements. Also we extend [10, Theorem 3.1 and Theorem 4.1] to quasi-clean rings in which every element is a sum of a potent element and a unit.

• Kyungpook Mathematical Journal
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• 제56권2호
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• pp.387-396
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• 2016

An $n{\times}n$ matrix A over a commutative ring is strongly clean provided that it can be written as the sum of an idempotent matrix and an invertible matrix that commute. Let R be an arbitrary commutative ring, and let $A(x){\in}M_n(R[[x]])$. We prove, in this note, that $A(x){\in}M_n(R[[x]])$ is strongly clean if and only if $A(0){\in}M_n(R)$ is strongly clean. Strongly clean matrices over quotient rings of power series are also determined.

• 한국환경보건학회지
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• 제35권4호
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• pp.322-333
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• 2009

In this study, a measurement method was evaluated for the determination of polycyclic aromatic hydrocarbons (PAHs) in the ambient atmosphere. PAHs were sampled by high-volume samplers, and were then analysed with a GC/MS system. Particulate PAHs were collected on $8"{\times}10"$ quartz fiber filter, while vapor phase PAHs were adsorbed on polyurethane foam (PUF). Target compounds included a total of 36 PAHs, which are known to be frequently detected in the urban atmosphere. It was not necessary to clean-up samples before samples were analyzed using GC/MS, and the overall performance of the method was tested by a variety of quality control and quality assurance schemes. It is generally known that the clean-up procedure can negatively affect the recovery of samples. Precision and accuracy was evaluated using SRM provided by US NIST, and the results were generally satisfactory and reliable. However, the GC/MS method appeared not to be adequate for 6-rings PAHs, such as coronene, due to its lower sensitivity. In addition, collection efficiencies for low molecular compounds, such as 2-rings PAHs, were poor because of the lower retention volume of the PUF adsorbent. As a result, it was concluded that the method based on high-volume sampling and GC/MS analysis can give very reliable data by simultaneous sampling of both particulate and vapor phases for 3-rings to 5-rings PAHs of environmental concern.