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

• Journal for History of Mathematics
• /
• v.10 no.2
• /
• pp.24-29
• /
• 1997

Derivation은 Lie group, Lie ring 그리고 Lie Algebra에서 정의되어 사용되며 발전하였으며 ring에서 일반화 되었다. 역시 prime ring에서 연구되어지는 derivation의 성질들은 prime near-ring에서 일반화 시키려고 하고 있다. 1957년 E. Posner는 prime ring에서 두 개의 derivation의 곱(함수합성)이 derivation이면 이들중 하나의 derivation이 0임을 밝혔다. 본 논문에서는 prime ring에서 derivation이 연구된 역사적인 배경을 소개하고 몇가지 성질을 찾는다. 즉, D. F를 prime ring R의 derivation들이라 할 때 정수 $n{\ge}1$에 대하여 $DF^n$=0이면 D=0이거나 또는 $F^{3n-1}$=0이고, $D^nF$=0이면 $D^{9n-7}$=0 이거나 또는 $F^2$=0 이다.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.4
• /
• pp.635-643
• /
• 2002

Our main goal is to show that if there exist Jordan derivations D and G on a noncommutative (n + 1)!-torsion free prime ring R such that $$D(x)x^n-x^nG(x)\in\ C(R)$$ for all $x\in\ R$, then we have D=0 and G=0. We also prove that if there exists a derivation D on a noncommutative 2-torsion free prime ring R such that the mapping $\chi$longrightarrow[aD($\chi$), $\chi$] is commuting on R, then we have either a = 0 or D = 0.

• Communications of the Korean Mathematical Society
• /
• v.28 no.4
• /
• pp.767-782
• /
• 2013

We deal with the Hyers-Ulam stability problem of linear mappings from a vector space into a Banach one with respect to the following functional equation: $$f\(\frac{-x+y}{3}\)+f\(\frac{x-3z}{3}\)+f\(\frac{3x-y+3z}{3}\)=f(x)$$. We then combine this equation with other ones and establish the Hyers-Ulam stability of several kinds of linear mappings, among which the algebra (*-) homomorphisms, the derivations, the multipliers and others. We thus repair and improve some previous assertions in the literature.

• Communications of the Korean Mathematical Society
• /
• v.32 no.3
• /
• pp.495-502
• /
• 2017

Let R be a 2-torsion free prime ring and J be a nonzero Jordan ideal of R. Let F and G be two generalized derivations with associated derivations f and g, respectively. Our main result in this paper shows that if F(x)x - xG(x) = 0 for all $x{\in}J$, then R is commutative and F = G or G is a left multiplier and F = G + f. This result with its consequences generalize some recent results due to El-Soufi and Aboubakr in which they assumed that the Jordan ideal J is also a subring of R.

• Kyungpook Mathematical Journal
• /
• v.45 no.2
• /
• pp.249-254
• /
• 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 $({\sigma},{\tau})-derivation$.

• Journal of the Chungcheong Mathematical Society
• /
• v.8 no.1
• /
• pp.37-44
• /
• 1995

In this paper we show that every left derivation on a semiprime Banach algebra A is a derivation which maps A into the intersection of the center of A and the Jacobson radical of A, and hence every left derivation on a semisimple Banach algebra is zero.

• Communications of the Korean Mathematical Society
• /
• v.37 no.1
• /
• pp.57-64
• /
• 2022

In this paper, we show, among others, that if A is a Banach algebra satisfying a functional identity involving a b-generalized derivation F on A, under some mild conditions, is of the form F(x) = ax for all x ∈ R, where a ∈ Q_r, a right Martindale quotient ring of A.

• Bulletin of the Korean Mathematical Society
• /
• v.61 no.2
• /
• pp.335-353
• /
• 2024

The objective of this paper is to study some central identities involving generalized derivations and anti-automorphisms in prime rings. Using the tools of the theory of functional identities, several known results have been generalized as well as improved.

• Communications of the Korean Mathematical Society
• /
• v.23 no.4
• /
• pp.469-477
• /
• 2008

We introduce the notion of two-sided $\Gamma$-$\alpha$-derivation of a $\Gamma$-near-ring and give some generalizations of [1, 2].