• East Asian mathematical journal
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• 제5권2호
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• pp.191-198
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• 1989

It is shown that a near-ring N which has no nonzero nilpotent elements is a partially ordered set where $x{\leq}y$ if and only if $yx=x^2$. Also it is shown that $(N,{\leq})$ is infinitely distributive for central elements that is $r(supx_i)=sup(rx_i)$ for every central element r of N and any subset $\{x_i\}$ of N. By using some lemmas we showed that a near-ring without nilpotent elements is isomorphic to a direct product of near-fields if and only if N is hyperatomic and orthogonally complete under the condition that every idempotent of N is central.

• 대한수학회보
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• 제48권2호
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• pp.277-290
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• 2011

We show that the ${\theta}$-prime radical of a ring R is the set of all strongly ${\theta}$-nilpotent elements in R, where ${\theta}$ is an automorphism of R. We observe some conditions under which the ${\theta}$-prime radical of coincides with the prime radical of R. Moreover we characterize elements in prime radicals of skew Laurent polynomial rings, studying (${\theta}$, ${\theta}^{-1}$)-(semi)primeness of ideals of R.

• 대한수학회보
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• 제49권2호
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• pp.353-358
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• 2012

In this paper, we focus on pseudo-Riemannian associative fermionic Novikov algebras. We prove that the underlying Lie algebras of pseudo-Riemannian associative fermionic Novikov algebras are 2-step nilpotent and that pseudo-Riemannian associative fermionic Novikov algebras are 3-step nilpotent. Moreover, we construct a pseudo-Riemannian associative fermionic Novikov algebra in dimension 14, which is not a Novikov algebra. It implies that the inverse proposition of Corollary 2 in the paper "Pseudo-Riemannian Novikov algebras" [J. Phys. A: Math. Theor. 41 (2008), 315207] does not hold.

• 대한수학회지
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• 제47권4호
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• pp.719-733
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• 2010

We observe a structure on the products of coefficients of nilpotent polynomials, introducing the concept of n-semi-Armendariz that is a generalization of Armendariz rings. We first obtain a classification of reduced rings, proving that a ring R is reduced if and only if the n by n upper triangular matrix ring over R is n-semi-Armendariz. It is shown that n-semi-Armendariz rings need not be (n+1)-semi-Armendariz and vice versa. We prove that a ring R is n-semi-Armendariz if and only if so is the polynomial ring over R. We next study interesting properties and useful examples of n-semi-Armendariz rings, constructing various kinds of counterexamples in the process.

• 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.

• 대한수학회보
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• 제54권2호
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• pp.455-461
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• 2017

The Lie algebra analogue of Schur's result which is proved by Moneyhun in 1994, states that if L is a Lie algebra such that dimL/Z(L) = n, then $dimL_{(2)}={\frac{1}{2}}n(n-1)-s$ for some non-negative integer s. In the present paper, we determine the structure of central factor (for s = 0) and the factor Lie algebra $L/Z_2(L)$ (for all $s{\geq}0$) of a finite dimensional nilpotent Lie algebra L, with n-dimensional central factor. Furthermore, by using the concept of n-isoclinism, we discuss an upper bound for the dimension of $L/Z_n(L)$ in terms of $dimL_{(n+1)}$, when the factor Lie algebra $L/Z_n(L)$ is finitely generated and $n{\geq}1$.

• 대한수학회논문집
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• 제33권3호
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• pp.751-757
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• 2018

A ring is nil-clean if every element is a sum of a nilpotent and an idempotent, and a ring is involution-clean if every element is a sum of an involution and an idempotent. In this paper, a description of nil-clean rings of nilpotency index at most 2 is obtained, and is applied to improve a known result on involution-clean rings.

• Korean Journal of Mathematics
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• 제24권3호
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• pp.307-317
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• 2016

Let R be a ring. It is well-known that R is NI if and only if ${\sum}^n_{i=0}Ra_i$ is a nil ideal of R whenever a polynomial ${\sum}^n_{i=0}a_ix^i$ is nilpotent, where x is an indeterminate over R. We consider a condition which is similar to the preceding one: ${\sum}^n_{i=0}Ra_iR$ contains a nonzero nil ideal of R whenever ${\sum}^n_{i=0}a_ix^i$ over R is nilpotent. A ring will be said to be quasi-NI if it satises this condition. The structure of quasi-NI rings is observed, and various examples are given to situations which raised naturally in the process.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제21권4호
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• pp.237-246
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• 2014

Let M be a prime ${\Gamma}$-ring and let d be a derivation of M. If there exists a fixed integer n such that $(d(x){\alpha})^nd(x)=0$ for all $x{\in}M$ and ${\alpha}{\in}{\Gamma}$, then we prove that d(x) = 0 for all $x{\in}M$. This result can be extended to semiprime ${\Gamma}$-rings.

• Korean Journal of Mathematics
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• 제21권4호
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• pp.421-428
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• 2013

We observe the basic structure of the products of coefficients of nilpotent (left) polynomials in skew polynomial rings. This study consists of a process to extend a well-known result for semi-Armendariz rings. We introduce the concept of ${\alpha}$-skew n-semi-Armendariz ring, where ${\alpha}$ is a ring endomorphism. We prove that a ring R is ${\alpha}$-rigid if and only if the n by n upper triangular matrix ring over R is $\bar{\alpha}$-skew n-semi-Armendariz. This result are applicable to several known results.