• 대한수학회논문집
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• 제14권1호
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• pp.69-74
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• 1999

In [2], the author discusses derivations of A(p,q) when A(p,q) is prime and p,q $\in$ A satisfy the condition A=Ap+Aq+R where R is a subspace of Z(A). In this paper, we consider a general-ization of Theorem 1 in [2] for the semiprime case of A(p,q).

• 호남수학학술지
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• 제36권3호
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• pp.505-517
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• 2014

We prove that every Jordan higher left (right) centralizer on a 2-torsion free semiprime ring is a higher left (right) centralizer which is to generalize the result of Zalar [18].

• East Asian mathematical journal
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• 제36권1호
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• pp.101-114
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• 2020

In this paper, we investigate commutativity of semiprime rings with a derivation which is strongly commutativity preserving and acts as a homomorphism or as an anti-homomorphism on a nonzero Lie ideal.

• East Asian mathematical journal
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• 제23권2호
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• pp.197-206
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• 2007

The purpose of this paper is to show some results of M. Bresar and J. Vukman concerning orthogonal derivations of semiprime rings in [3] for (${\sigma},\;{\tau}$)-derivations and generalized (${\sigma},\;{\tau}$)-derivations in $\Gamma$-rings.

• 대한수학회논문집
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• 제32권2호
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• pp.277-285
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• 2017

In the various module generalizations of the concepts of prime (semiprime) for a ring, the question "when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If R is Morita equivalent to a right pre-duo ring (e.g., if R is commutative) then weakly compressible R-modules are precisely subdirect products of prime R-modules if and only if dim(R) = 0 and R/N(R) is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible R-modules is an enveloping for Mod-R. Some related conditions are also investigated.

• 대한수학회지
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• 제47권5호
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• pp.879-897
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• 2010

Y. Hirano introduced the concept of a quasi-Armendariz ring which extends both Armendariz rings and semiprime rings. A ring R is called quasi-Armendariz if $a_iRb_j$ = 0 for each i, j whenever polynomials $f(x)\;=\;\sum_{i=0}^ma_ix^i$, $g(x)\;=\;\sum_{j=0}^mb_jx^j\;{\in}\;R[x]$ satisfy f(x)R[x]g(x) = 0. In this paper, we first extend the quasi-Armendariz property of semiprime rings to the skew polynomial rings, that is, we show that if R is a semiprime ring with an epimorphism $\sigma$, then f(x)R[x; $\sigma$]g(x) = 0 implies $a_iR{\sigma}^{i+k}(b_j)=0$ for any integer k $\geq$ 0 and i, j, where $f(x)\;=\;\sum_{i=0}^ma_ix^i$, $g(x)\;=\;\sum_{j=0}^mb_jx^j\;{\in}\;R[x,\;{\sigma}]$. Moreover, we extend this property to the skew monoid rings, the Ore extensions of several types, and skew power series ring, etc. Next we define $\sigma$-skew quasi-Armendariz rings for an endomorphism $\sigma$ of a ring R. Then we study several extensions of $\sigma$-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and $\sigma$-skew Armendariz rings.

• Journal of applied mathematics & informatics
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• 제36권1_2호
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• pp.81-92
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• 2018

In this article we study two Multiplicative (generalized)- derivations ${\mathcal{G}}$ and ${\mathcal{H}}$ that satisfying certain conditions in semiprime rings and tried to find out some information about the associated maps. Moreover, an example is given to demonstrate that the semiprimeness imposed on the hypothesis of the various results is essential.

• 대한수학회보
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• 제51권1호
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• pp.207-211
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• 2014

The aim of this paper is to prove the next result. Let n > 1 be an integer and let R be a n!-torsion free semiprime ring. Suppose that f : R ${\rightarrow}$ R is an additive mapping satisfying the relation [f(x), $x^n$] = 0 for all $x{\in}R$. Then f is commuting on R.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제14권4호
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• pp.271-278
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• 2007

Let R be a 2-torsion free semiprime ring. Suppose that there exists a 4-permuting 4-derivation ${\Delta}:R{\times}R{\times}R{\times}R{\rightarrow}R$ such that the trace is centralizing on R. Then the trace of ${\Delta}$ is commuting on R. In particular, if R is a 3!-torsion free prime ring and ${\Delta}$ is nonzero under the same condition, then R is commutative.

• 충청수학회지
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• 제22권3호
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• pp.451-458
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• 2009

Let $n{\geq}2$ be a fixed positive integer and let R be a noncommutative n!-torsion free semiprime ring. Suppose that there exists a symmetric n-derivation $\Delta$ : $R^{n}{\rightarrow}R$ such that the trace of $\Delta$ is centralizing on R. Then the trace is commuting on R. If R is a n!-torsion free prime ring and $\Delta{\neq}0$ under the same condition. Then R is commutative.