• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1281-1293
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• 2023

The purpose of this paper is to study the relationship between the structure of a factor ring R/P and the behavior of some derivations of R. More precisely, we establish a connection between the commutativity of R/P and derivations of R satisfying specific identities involving the prime ideal P. Moreover, we provide an example to show that our results cannot be extended to semi-prime ideals.

• Journal of applied mathematics & informatics
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• v.42 no.3
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• pp.539-548
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• 2024

Let R be a ring and P be a prime ideal of R. A mapping d : R → R is called a multiplicative derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R. In this paper, our main motive is to obtain the well-known theorem due to Posner in the ring R/P for generalized derivations associated with a multiplicative derivation defined by an additive mapping F : R → R such that F(xy) = F(x)y + xd(y), where d : R → R is a multiplicative derivation not necessarily additive. This article discusses the use of generalized derivations associated with a multiplicative derivation to investigate the commutativity of the quotient ring R/P.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.249-254
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• 2005

Let N be a prime left near-ring with multiplicative center Z and f be a generalized $({\sigma},{\tau})-derivation$ associated with d. We prove commutativity theorems in prime near- rings with generalized $({\sigma},{\tau})-derivation$.

• Kyungpook Mathematical Journal
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• v.27 no.2
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• pp.187-189
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• 1987

In the present paper a result [10] of the authors has been generalized as follows: Let l, m, n be fixed positive integers and R be a semi prime ring in which $[(xy)^l,(xy)^m-(yx)^n]=0$ for all $x,y{\in}R$, then R is commutative.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.541-544
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• 1995

In [1], Ahmad has given a characterization of prime dual ideals in bounded commutative BCK-algebras. The aime of this paper is to show that Theorem of [1] holds without the commutativity.

• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.1-9
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• 2010

In this note, we introduce a permuting 3-derivation in nearrings and investigate the conditions for a near-ring to be a commutative ring.

• Korean Journal of Mathematics
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• v.16 no.3
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• pp.281-290
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• 2008

The purpose of this paper, GM modules are defined as a generalization of AGR rings. Also, from the faithful GM-property, we get a commutativity of rings. Finally, for a nearring R, we will introduce MR groups, and also derive some properties of GM modules and MR groups.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.203-217
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• 2009

In this note, we introduce a symmetric generalized 3-derivation in near-rings and investigate some conditions for a nearring to be a commutative ring.

• Communications of the Korean Mathematical Society
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• v.26 no.3
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• pp.445-454
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• 2011

In this paper, we extend some well known results concerning generalized derivations of prime rings to a generalized (${\sigma}$, ${\tau}$)-derivation.

• Kyungpook Mathematical Journal
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• v.57 no.1
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• pp.125-132
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• 2017

We obtain commutativity-free characterizations of higher derivations on a unital Banach algebra that map into its Jacobson radical. We also investigate the spectral boundedness of generalized higher derivations.