• 대한수학회지
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• 제56권5호
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• pp.1159-1172
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• 2019

A semigroup S is called a weakly abundant semigroup if its every $\tilde{\mathcal{L}}$-class and every $\tilde{\mathcal{R}}$-class contains an idempotent. Our purpose is to study an analogue of orthodox semigroups in the class of weakly abundant semigroups. Such an analogue is called a left quasi-abundant semigroup, which is a weakly abundant semigroup with a left quasi-normal band of idempotents and having the congruence condition (C). To build our main structure theorem for left quasi-abundant semigroups, we first give a sufficient and necessary condition of the idempotent set E(S) of a weakly abundant semigroup S being a left quasi-normal band. And then we construct a left quasi-abundant semigroup in terms of weak spined products. Such a result is a generalisation of that of Guo and Shum for left semi-perfect abundant semigroups. In addition, we consider a type Q semigroup which is a left quasi-abundant semigroup having the PC condition.

• Kyungpook Mathematical Journal
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• 제55권4호
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• pp.885-892
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• 2015

In this paper, we mainly obtain the following assertions: (1) If T is a quasi-*-n-paranormal operator, then T is finite and simply polaroid. (2) If T or $T^*$ is a quasi-*-n-paranormal operator, then Weyl's theorem holds for f(T), where f is an analytic function on ${\sigma}(T)$ and is not constant on each connected component of the open set U containing ${\sigma}(T)$. (3) If E is the Riesz idempotent for a nonzero isolated point ${\lambda}$ of the spectrum of a quasi-*-n-paranormal operator, then E is self-adjoint and $EH=N(T-{\lambda})=N(T-{\lambda})^*$.

• 대한수학회보
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• 제27권2호
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• pp.113-120
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• 1990

In this paper we introduce S(X, $x_{0}$) which is a generalization of Ellis group G(X, $x_{0}$), and S-sets in S(X, $x_{0}$). In particular we cind the sufficient condition for the group A(I) of all automorphisms of I and K=Iu to be isomorphic, where I is a minimal right ideal and u is an idempotent of I.f I.

• Korean Journal of Mathematics
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• 제9권2호
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• pp.99-104
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• 2001

We give the characterization of left(right) semi-regular $po$-semigroups and $Ve$-semigroups.

• 대한수학회논문집
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• 제11권1호
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• pp.63-70
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• 1996

Let K be a field of characteristic 0 and G a Klein's four group. We find the idempotent elements and units of the group ring KG by using the basic group table matrix of G.

• 대한수학회논문집
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• 제35권1호
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• pp.117-124
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• 2020

In 2013, Diesl defined a nil clean ring as a ring of which all elements can be expressed as the sum of an idempotent and a nilpotent. Furthermore, in 2017, Y. Zhou, S. Sahinkaya, G. Tang studied nil clean group rings, finding both necessary condition and sufficient condition for a group ring to be a nil clean ring. We have proposed a necessary and sufficient condition for a group ring to be a uniquely nil clean ring. Additionally, we provided theorems for general nil clean group rings, and some examples of trivial-center groups of which group ring is not nil clean over any strongly nil clean rings.

• 대한수학회논문집
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• 제32권3호
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• pp.661-676
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• 2017

Let T be a bounded linear operator acting on a complex Hilbert space ${\mathfrak{H}}$. In this paper we introduce the class, denoted ${\mathcal{Q}}(A(k),m)$, of operators satisfying $T^{m{\ast}}(T^{\ast}{\mid}T{\mid}^{2k}T)^{1/(k+1)}T^m{\geq}T^{{\ast}m}{\mid}T{\mid}^2T^m$, where m is a positive integer and k is a positive real number and we prove basic structural properties of these operators. Using these results, we prove that if P is the Riesz idempotent for isolated point ${\lambda}$ of the spectrum of $T{\in}{\mathcal{Q}}(A(k),m)$, then P is self-adjoint, and we give a necessary and sufficient condition for $T{\otimes}S$ to be in ${\mathcal{Q}}(A(k),m)$ when T and S are both non-zero operators. Moreover, we characterize the quasinilpotent part $H_0(T-{\lambda})$ of class A(k) operator.

• 대한수학회논문집
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• 제21권2호
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• pp.227-236
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• 2006

Several authors find all the derivations of an algebra [1], [3], [7]. A Weyl type non-associative algebra and its sub algebra are defined in the paper [2], [3], [10]. All the derivations of the non-associative algebra $\overline{WN_{0,0,s1}$ is found in this paper [4]. We find all the derivations of the non-associative algebra $\overline{WN_{0,s,01}$ in this paper [5].

• 대한수학회논문집
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• 제33권1호
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• pp.1-12
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• 2018

We introduce the notion of ordered intra k-regular semirings, characterize them using their ordered k-ideals and prove that an ordered semiring S is both ordered k-regular and ordered intra k-regular if and only if every ordered quasi k-ideal or every ordered k-bi-ideal of S is ordered k-idempotent.

• 대한수학회지
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• 제51권3호
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• pp.463-472
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• 2014

Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R and $S_{\ell}$(R) (resp. $S_r$(R)) be the set of all left (resp. right) semicentral idempotents in R. In this paper, the following are investigated: (1) $e{\in}S_{\ell}(R)$ (resp. $e{\in}S_r(R)$) if and only if re=ere (resp. er=ere) for all nilpotent elements $r{\in}R$ if and only if $fe{\in}I(R)$ (resp. $ef{\in}I(R)$) for all $f{\in}I(R)$ if and only if fe=efe (resp. ef=efe) for all $f{\in}I(R)$ if and only if fe=efe (resp. ef=efe) for all $f{\in}I(R)$ which are isomorphic to e if and only if $(fe)^n=(efe)^n$ (resp. $(ef)^n=(efe)^n$) for all $f{\in}I(R)$ which are isomorphic to e where n is some positive integer; (2) For a ring R having a complete set of centrally primitive idempotents, every nonzero left (resp. right) semicentral idempotent is a finite sum of orthogonal left (resp. right) semicentral primitive idempotents, and eRe has also a complete set of primitive idempotents for any $0{\neq}e{\in}S_{\ell}(R)$ (resp. 0$0{\neq}e{\in}S_r(R)$).