• 대한수학회보
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• 제54권1호
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• pp.51-69
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• 2017

We, in this paper, study the commutativity of skew polynomials at zero as a generalization of an ${\alpha}-rigid$ ring, introducing the concept of strongly skew reversibility. A ring R is be said to be strongly ${\alpha}-skew$ reversible if the skew polynomial ring $R[x;{\alpha}]$ is reversible. We examine some characterizations and extensions of strongly ${\alpha}-skew$ reversible rings in relation with several ring theoretic properties which have roles in ring theory.

• 대한수학회논문집
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• 제26권3호
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• pp.339-348
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• 2011

A ring R is called symmetric, if abc = 0 implies acb = 0 for a, b, c ${\in}$ R. An ideal I of a ring R is called symmetric (resp. radically-symmetric) if R=I (resp. R/$\sqrt{I}$) is a symmetric ring. We first show that symmetric ideals and ideals which have the insertion of factors property are radically-symmetric. We next show that if R is a semicommutative ring, then $T_n$(R) and R[x]=($x^n$) are radically-symmetric, where ($x^n$) is the ideal of R[x] generated by $x^n$. Also we give some examples of radically-symmetric ideals which are not symmetric. Connections between symmetric ideals of R and related ideals of some ring extensions are also shown. In particular we show that if R is a symmetric (or semicommutative) (${\alpha}$, ${\delta}$)-compatible ring, then R[x; ${\alpha}$, ${\delta}$] is a radically-symmetric ring. As a corollary we obtain a generalization of [13].

• Korean Journal of Mathematics
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• 제18권2호
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• pp.119-132
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• 2010

P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for $a,b{\in}R$. In this paper, we study an extension of a reversible ring with its endomorphism. An endomorphism ${\alpha}$ of a ring R is called strong right (resp., left) reversible if whenever $a{\alpha}(b)=0$ (resp., ${\alpha}(a)b=0$) for $a,b{\in}R$, ba = 0. A ring R is called strong right (resp., left) ${\alpha}$-reversible if there exists a strong right (resp., left) reversible endomorphism ${\alpha}$ of R, and the ring R is called strong ${\alpha}$-reversible if R is both strong left and right ${\alpha}$-reversible. We investigate characterizations of strong ${\alpha}$-reversible rings and their related properties including extensions. In particular, we show that every semiprime and strong ${\alpha}$-reversible ring is ${\alpha}$-rigid and that for an ${\alpha}$-skew Armendariz ring R, the ring R is reversible and strong ${\alpha}$-reversible if and only if the skew polynomial ring $R[x;{\alpha}]$ of R is reversible.

• 대한화학회지
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• 제37권3호
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• pp.344-350
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• 1993

25,26,27,28-테트라아세트오키시[4]에렌·일수화물은 사방정계의 Pbca의 공간군을 갖고 있으며 a = 14.979(4), b = 15.154(4), c = 27.890(3) ${\AA}$, Z = 8, V = 6330.6 ${\AA}^{-3}$, D$_c$ = 1.28 $g{\cdot}cm^{-3}$, ${\lambda}$(Mo K${\alpha}$) = 0.71069 ${\AA}$, ${\mu}$ = 0.86 cm$^{-1}$, F(000) = 2600이고, 1.0 ${\sigma}$(I)보다 큰 강도를 가진 3376갱의 관측된 회절반점에 대하여 최종 R값은 0.069이다. 직접적에 의하여 구조를 풀었으며, 계단식 대각최소자승법에 의하여 정밀화하여, 모든 C-H 결합길이 (=0.96 ${\AA}$) 및 메칠기와 메칠렌기는 이상적인 기하학적 구조에 맞추어 계산하였다. 큰 고리는 1,3 alternate conformation을 하고 있으며 메칠렌기의 평균 평면으로부터 벤젠고리는 각각 110.7, 68.4, 113.7 및 60.8$^{\circ}$를 이루고 있으며, 4개의 각 아세트오키시기는 그들 자신의 벤젠고리와 각각 68.2, 97.6, 78.9 및 71.3$^{\circ}$의 각을 만들고 있다. 반대편에 마주 위치한 벤젠고리 (1)과 (3)은 135.6$^{\circ}$, (2)과 (4)는 135.2$^{\circ}$의 상대적 각을 이루고 있다. 물분자는 메칠렌기를 포한하고 잇는 큰고리의 평면의 z-좌표와 거의 같은 높이에 있으며, 분자내의 O(8)부터 2.942(5) ${\AA}$, 타분자의 O(2)(1/2-x, -1/2＋y, z)부터는 2.901 ${\AA}$의 거리에 있으며, 가장 짧은 분자간 거리는 O(4) 와 C(3)(1/2＋x, 1/2-y,-z)의 3.193 ${\AA}$이다.