• East Asian mathematical journal
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• 제15권2호
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• pp.165-176
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• 1999

After the derivation was defined in [19] by Posner a lot of researchers studied the derivations in ring theory in different manners such as in [2], [4], [5], ..., etc. Furthermore, many researches followed the definition of the generalized derivation([3], [6], [7], ..., etc.). Finally, Maksa defined a symmetric bi-derivation and many researches have been done in ring theory by using this definition. In this work, defining a symmetric bi-$\alpha$-derivation, we study the mentioned researches above in the light of this new concept.

• East Asian mathematical journal
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• 제14권2호
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• pp.259-268
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• 1998

In this paper, for a commutative ring R with an identity, considering the endomorphism ring $End_R$(M) of left R-module $_RM$ which is (quasi-)injective or (quasi-)projective, some discriminations of prime endomorphism were found as follows: each epimorphism with the irreducible(or simple) kernel on a (quasi-)injective module and each monomorphism with maximal image on a (quasi-)projective module are prime. It was shown that for a field F, any given square matrix in $Mat_{n{\times}n}$(F) with maximal image and irreducible kernel is a prime matrix, furthermore, any given matrix in $Mat_{n{\times}n}$(F) for any field F can be factored into a product of prime matrices.

• Korean Journal of Mathematics
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• 제27권1호
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• pp.141-150
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• 2019

In this article we investigate the structure of rings in which lower nilradicals coincide with upper nilradicals. Such rings shall be said to be quasi-2-primal. It is shown first that the $K{\ddot{o}}the^{\prime}s$ conjecture holds for quasi-2-primal rings. So the results in this article may provide interesting and useful information to the study of nilradicals in various situations. In the procedure we study the structure of quasi-2-primal rings, and observe various kinds of quasi-2-primal rings which do roles in ring theory.

• 대한수학회보
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• 제45권4호
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• pp.621-629
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• 2008

Let R be a prime ring of characteristic different from 2, C the extended centroid of R, and $\delta$ a generalized derivations of R. If [[$\delta(x)$, x], $\delta(x)$] = 0 for all $x\;{\in}\;R$ then either R is commutative or $\delta(x)\;=\;ax$ for all $x\;{\in}\;R$ and some $a\;{\in}\;C$. We also obtain some related result in case R is a Banach algebra and $\delta$ is either continuous or spectrally bounded.

• East Asian mathematical journal
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• 제22권2호
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• pp.239-248
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• 2006

Let G be a group acting on a ring R. We will define the group action of G on an intuitionsitic fuzzy set of R. We will introduce intuitionistic fuzzy G-prime ideals of a ring and we will prove that every intuitionistic fuzzy G-prime ideal is the largest G-invariant intuitionistic fuzzy ideal of R contained in the intuitionistic fuzzy prime ideal which is uniquely determined up to G-orbits.

• Journal of applied mathematics & informatics
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• 제6권1호
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• pp.239-246
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• 1999

The purpose of this paper is to prove the following result: Let R be a 2-torsion free noncommutative semi-prime ring and D:RlongrightarrowR a derivation. Suppose that $[[D(\chi),\chi],\chi]\in$ Z(R) holds for all $\chi \in R$. Then D is commuting on R.

• Kyungpook Mathematical Journal
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• 제58권3호
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• pp.481-488
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• 2018

Let I be a ${\ast}$-ideal on a 2-torsion free ${\ast}$-prime ring and $F:R{\rightarrow}R$ a generalized derivation with an associated derivation $d:R{\rightarrow}R$. The aim of this paper is to explore the condition under which generalized derivation F becomes a left centralizer i.e., associated derivation d becomes a trivial map (i.e., zero map) on R.

• 대한수학회보
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• 제46권5호
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• pp.857-866
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• 2009

Let n be a fixed positive integer, R be a 2n!-torsion free prime ring and $\mu$, $\nu$ be a pair of generalized derivations on R. If < $\mu^2(x)+\nu(x),\;x^n$ > = 0 for all x $\in$ R, then $\mu$ and $\nu$ are either left multipliers or right multipliers. Let n be a fixed positive integer, R be a noncommutative 2n!-torsion free prime ring with the center $C_R$ and d, g be a pair of derivations on R. If < $d^2(x)+g(x)$, $x^n$ > $\in$ $C_R$ for all x $\in$ R, then d = g = 0. Then we apply these purely algebraic techniques to obtain several range inclusion results of pair of (generalized-)derivations on a Banach algebra.

• 대한수학회보
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• 제55권2호
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• pp.573-586
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• 2018

Let R be a noncommutative prime ring of characteristic different from 2, Q be its maximal right ring of quotients and C be its extended centroid. Suppose that $f(x_1,{\ldots},x_n)$ be a noncentral multilinear polynomial over $C,b{\in}Q,F$ a b-generalized derivation of R and d is a nonzero derivation of R such that d([F(f(r)), f(r)]) = 0 for all $r=(r_1,{\ldots},r_n){\in}R^n$. Then one of the following holds: (1) there exists ${\lambda}{\in}C$ such that $F(x)={\lambda}x$ for all $x{\in}R$; (2) there exist ${\lambda}{\in}C$ and $p{\in}Q$ such that $F(x)={\lambda}x+px+xp$ for all $x{\in}R$ with $f(x_1,{\ldots},x_n)^2$ is central valued in R.

• 대한수학회보
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• 제59권4호
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• pp.853-868
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• 2022

We study the structure of right regular commutators, and call a ring R strongly C-regular if ab - ba ∈ (ab - ba)²R for any a, b ∈ R. We first prove that a noncommutative strongly C-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly C-regular if and only if it is Abelian C-regular (from which we infer that strong C-regularity is left-right symmetric). It is proved that for a strongly C-regular ring R, (i) if R/W(R) is commutative, then R is commutative; and (ii) every prime factor ring of R is either a commutative domain or a noncommutative division ring, where W(R) is the Wedderburn radical of R.