• Kyungpook Mathematical Journal
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• 제19권2호
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• pp.205-211
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• 1979

In this paper we continue the study of formation radical (F-radical) classes initiated in [3]. Hereditary and stronger properties of F-radical classes are discussed by giving construction for lower hereditary, lower stronger and lower strongly hereditary F-radical classes containing a given class M. It is shown that the Baer F-radical B is the lower strongly hereditary F-radical class containing the class of all nilpotent ideals and it is the upper radical class with $\{(I,\;N){\mid}N{\in}C,\;N\;is\;prime\}{\subset}SB$ where SB denotes the semisimple F-radical class of B and C is an arbitrary but fixed class of homomorphically closed near-rings. The existence of a largest F-radical class contained in a given class is examined using the concept of complementary F-radical introduced by Scott [5].

• Journal of applied mathematics & informatics
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• 제8권3호
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• pp.855-867
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• 2001

This note we give a construction of a quotient ring $R/{\mu}$ induced via a fuzzy ideal ${\mu}$ in a ring R. The Fuzzy First, Second and Third Isomorphism Theorems are established. For some applications of this construction of quotient rings, we show that if ${\mu}$ is a fuzzy ideal of a commutative ring R, then $\mu$ is prime (resp. $R/{\mu}$ is a field, every zero divisor in $R/{\mu}$ is nilpotent). Moreover we give a simpler characterization of fuzzy maximal ideal of a ring.

• East Asian mathematical journal
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• 제18권2호
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• pp.195-203
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• 2002

Let R be a prime ring with characteristic not two and U is a nonzero left ideal of R which contains no nonzero nilpotent right ideal as a ring. For a $(\sigma,\tau)$-derivation d : R$\rightarrow$R, we prove the following results: (1) If d is an endomorphism on R then d=0. (2) If d is an anti-endomorphism on R then d=0. (3) If d(xy)=d(yx), for all x, y$\in$R then R is commutative. (4) If d is an homomorphism or anti-homomorphism on U then d=0.

• 대한수학회지
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• 제51권6호
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• pp.1251-1267
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• 2014

The semicommutative property of rings was introduced initially by Bell, and has done important roles in noncommutative ring theory. This concept was generalized to one of nil-semicommutative by Chen. We first study some basic properties of nil-semicommutative rings. We next investigate the structure of Ore extensions when upper nilradicals are ${\sigma}$-rigid ${\delta}$-ideals, examining the nil-semicommutative ring property of Ore extensions and skew power series rings, where ${\sigma}$ is a ring endomorphism and ${\delta}$ is a ${\sigma}$-derivation.

• 대한수학회지
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• 제57권3호
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• pp.707-719
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• 2020

In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a (2n + 1)-dimensional Sasakian manifold admits a weakly Einstein metric, then its scalar curvature s satisfies -6 ⩽ s ⩽ 6 for n = 1 and -2n(2n + 1) ${\frac{4n^2-4n+3}{4n^2-4n-1}}$ ⩽ s ⩽ 2n(2n + 1) for n ⩾ 2. Secondly, for a (2n + 1)-dimensional weakly Einstein contact metric (κ, μ)-manifold with κ < 1, we prove that it is flat or is locally isomorphic to the Lie group SU(2), SL(2), or E(1, 1) for n = 1 and that for n ⩾ 2 there are no weakly Einstein metrics on contact metric (κ, μ)-manifolds with 0 < κ < 1. For κ < 0, we get a classification of weakly Einstein contact metric (κ, μ)-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic (κ, μ)-manifold with κ < 0 is locally isomorphic to a solvable non-nilpotent Lie group.

• Korean Journal of Mathematics
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• 제26권2호
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• pp.175-189
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• 2018

In this paper, at first we generalize the notion of algebra over a field. A ${\Gamma}$-algebra is an algebraic structure consisting of a vector space V, a groupoid ${\Gamma}$ together with a map from $V{\times}{\Gamma}{\times}V$ to V. Then, on every associative ${\Gamma}$-algebra V and for every ${\alpha}{{\in}}{\Gamma}$ we construct an ${\alpha}$-Lie algebra. Also, we discuss some properties about ${\Gamma}$-Lie algebras when V and ${\Gamma}$ are the sets of $m{\times}n$ and $n{\times}m$ matrices over a field F respectively. Finally, we define the notions of ${\alpha}$-derivation, ${\alpha}$-representation, ${\alpha}$-nilpotency and prove Engel theorem in this case.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제4권1호
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• pp.61-69
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• 1997

In this paper, we study the dynamics of a two-parameter unfolding system $\chi$' = y, y' = $\beta$y＋$\alpha$f($\chi\alpha\pm\chiy$＋yg($\chi$), where f($\chi$,$\alpha$) is a second order polynomial in $\chi$ and g($\chi$) is strictly nonlinear in $\chi$. We show that the higher order term yg($\chi$) in the system does not change qulitative structure of the Hopf bifurcations near the fixed points for small $\alpha$ and $\beta$ if the nontrivial fixed point approaches to the origin as $\alpha$ approaches zero.

• 충청수학회지
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• 제14권1호
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• pp.21-28
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• 2001

In this paper we show that the following results remain valid for arbitrary Jordan derivations as well: Let d be a derivation of a complex Banach algebra A. If $d^2(x){\in}rad(A)$ for all $x{\in}A$, then we have $d(A){\subseteq}rad(A)$ ([5, p. 243]), and in a case when A is unital, $d(A){\subseteq}rad(A)$ if and only if sup{$r(z^{-1}d(z)){\mid}z{\in}A$ invertible} < ${\infty}$([3]), where rad(A) stands for the Jacobson radical of A, and r(${\cdot}$) for the spectral radius.

• 대한수학회지
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• 제58권6호
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• pp.1407-1419
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• 2021

The Iwasawa decomposition NAK of the Lie group G = SO₀(2, 1) with a left invariant metric produces Riemannian submersions G → N\G, G → A\G, G → K\G, and G → NA\G. For each of these, we calculate the curvature of the base space and the lifting of a simple closed curve to the total space G. Especially in the first case, the base space has a constant curvature 0; the holonomy displacement along a (null-homotopic) simple closed curve in the base space is determined only by the Euclidean area of the region surrounded by the curve.

• 대한수학회지
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• 제56권2호
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• pp.289-309
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• 2019

This paper is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commutative property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpotent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series.