• Bulletin of the Korean Mathematical Society
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• v.40 no.4
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• pp.545-551
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• 2003

We define subalgebras ${V_q}^{mZ{\times}nZ}\;of\;V_q\;where\;V_q$ are in the paper [4]. We show that the Lie algebra ${V_q}^{mZ{\times}nZ}$ is simple and maximally abelian decomposing. We may define a Lie algebra is maximally abelian decomposing, if it has a maximally abelian decomposition of it. The F-algebra automorphism group of the Laurent extension of the quantum plane is found in the paper [4], so we find the Lie automorphism group of ${V_q}^{mZ{\times}nZ}$ in this paper.

• East Asian mathematical journal
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• v.36 no.1
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• pp.73-79
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• 2020

A ring R is called generalized regular if for every nonzero x in R there exists y in R such that xy is a nonzero idempotent. In this paper, we observe some equivalent conditions for the generalized regular rings that are abelian in terms of idempotents, and we also investigate the primitivity of an idempotent for such a ring. By using the investigation, we characterize such a kind of rings that are noetherian by showing that an abelian generalized regular ring R is noetherian if and only if R is isomorphic to a direct product of finitely many division rings. We also observe some interesting consequences of our results.

• Honam Mathematical Journal
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• v.36 no.2
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• pp.417-424
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• 2014

Let A be an abelian variety defined over a number field K and p be a prime. Define ${\varphi}_i=(x^{p^i}-1)/(x^{p^{i-1}}-1)$. Let $A_{{\varphi}i}$ be the abelian variety defined over K associated to the polynomial ${\varphi}i$ and let Ш($A_{{\varphi}i}$) denote the Tate-Shafarevich groups of $A_{{\varphi}i}$ over K. In this paper assuming Ш(A/F) is finite, we compute [Ш($A_{{\varphi}1}$)][Ш($A_{{\varphi}2}$)]/[Ш($A_{{\varphi}1{\varphi}2}$)] in terms of K-rational points of $A_{{\varphi}i}$, $A_{{\varphi}1{\varphi}2}$ and their dual varieties, where [X] is the order of a finite abelian group X.

• International Journal of Computer Science & Network Security
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• v.21 no.8
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• pp.17-27
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• 2021

Non-abelian group based cryptosystems are a latest research inspiration, since they offer better security due to their non-abelian properties. In this paper, we propose a novel approach to non-abelian group based public-key cryptographic protocols using semidirect products of finite groups. An intractable problem of determining automorphisms and generating elements of a group is introduced as the underlying mathematical problem for the suggested protocols. Then, we show that the difficult problem of determining paths and cycles of Cayley graphs including Hamiltonian paths and cycles could be reduced to this intractable problem. The applicability of Hamiltonian paths, and in fact any random path in Cayley graphs in the above cryptographic schemes and an application of the same concept to two previous cryptographic protocols based on a Generalized Discrete Logarithm Problem is discussed. Moreover, an alternative method of improving the security is also presented.

• East Asian mathematical journal
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• v.37 no.1
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• pp.33-40
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• 2021

We discuss the condition that if ab = 0 for elements a, b in a ring R then aIb = 0 for some essential ideal I of R. A ring with such condition is called IEIP. We prove that a ring R is IEIP if and only if Dn(R) is IEIP for every n ≥ 2, where Dn(R) is the ring of n by n upper triangular matrices over R whose diagonals are equal. We construct an IEIP ring that is not Abelian and show that a well-known Abelian ring is not IEIP, noting that rings with the insertion-of-factors-property are Abelian.

• East Asian mathematical journal
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• v.25 no.4
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• pp.433-440
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• 2009

Let R be a ring with unity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will consider some group actions on X by G, the left (resp. right) regular action and the conjugate action. In this paper, by investigating these group actions we can have some results as follows: First, if E(R), the set of all nonzero nonunit idempotents of a unit-regular ring R, is commuting, then $o_{\ell}(x)\;=\;o_r(x)$, $o_c(x)\;=\;\{x\}$ for all $x\;{\in}\;X$ where $o_{\ell}(x)$ (resp. $o_r(x)$, $o_c(x)$) is the orbit of x under the left regular (resp. right regular, conjugate) action on X by G and R is abelian regular. Secondly, if R is a unit-regular ring with unity 1 such that G is a cyclic group and $2\;=\;1\;+\;1\;{\in}\;G$, then G is a finite group. Finally, if R is an abelian regular ring such that G is an abelian group, then R is a commutative ring.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.295-300
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• 2016

For the complex torus, there is the Riemann condition on the polarization for it to be an abelian variety. We extend this condition on the super torus for super abelian variety.

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.199-209
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• 2002

In this paper, we investigate the problem of stability of the quadratic functional equation f(x+y+z)+f(y-z)+f(y-z)+f(x-z) = 3f(x)+3f(y)+3f(z) on abelian group.

• Journal of applied mathematics & informatics
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• v.5 no.2
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• pp.393-402
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• 1998

In this paper we describe the results of computations to find symmetric presentations based on minimal conjugate generating pairs for the 32 non-abelian simple groups of order less than one million.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.763-769
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• 2007

We study the structure of abelian-by-nilpotent groups satisfying the maximal condition on infinite normal subgroups or the minimal condition on normal subgroups of infinite index.