• The Pure and Applied Mathematics
• /
• v.25 no.3
• /
• pp.181-191
• /
• 2018

In the present paper, we investigate the action of generalized derivation G associated with a derivation g in a (semi-) prime ring R satisfying $(G([x,y])-[G(x),y])^n=0$ for all x, $y{\in}I$, a nonzero ideal of R, where n is a fixed positive integer. Moreover, we also examine the above identity in Banach algebras.

• East Asian mathematical journal
• /
• v.21 no.1
• /
• pp.21-26
• /
• 2005

The purpose of this paper is to show that every generalized Jordan derivation of prime ring with characteristic not two is a generalized derivation on a nonzero Lie ideal U of R such that $u^2{\in}U\;for\;{\forall}u{\in}U$ which is a generalization of the well-known result of I. N. Herstein.

• Bulletin of the Korean Mathematical Society
• /
• v.57 no.3
• /
• pp.709-737
• /
• 2020

Let R be a prime ring of characteristic different from 2. Suppose that F, G, H and T are generalized derivations of R. Let U be the Utumi quotient ring of R and C be the center of U, called the extended centroid of R and let f(x₁, …, x_n) be a non central multilinear polynomial over C. If F(f(r₁, …, r_n))G(f(r₁, …, r_n)) - f(r₁, …, r_n)T(f(r₁, …, r_n)) = H(f(r₁, …, r_n)²) for all r₁, …, r_n ∈ R, then we describe all possible forms of F, G, H and T.

• Journal of applied mathematics & informatics
• /
• v.40 no.5_6
• /
• pp.915-923
• /
• 2022

The purpose of the present paper is to obtain commutativity of prime Γ-near-ring N with generalized derivations F and G with associated derivations d and h respectively satisfying one of the following conditions:(i) G([x, y]α = ±f(y)α(xoy)βγg(y), (ii) F(x)βG(y) = G(y)βF(x), for all x, y ∈ N, β ∈ Γ (iii) F(u)βG(v) = G(v)βF(u), for all u ∈ U, v ∈ V, β ∈ Γ,(iv) if 0 ≠ F(a) ∈ Z(N) for some a ∈ V such that F(x)αG(y) = G(y)αF(x) for all x ∈ V and y ∈ U, α ∈ Γ.

• Journal of the Korean Mathematical Society
• /
• v.47 no.3
• /
• pp.483-494
• /
• 2010

Let R be a non-commutative prime ring and I a non-zero left ideal of R. Let U be the left Utumi quotient ring of R and C be the center of U and k, m, n, r fixed positive integers. If there exists a generalized derivation g of R such that $[g(x^m)x^n,\;x^r]_k\;=\;0$ for all x $\in$ I, then there exists a $\in$ U such that g(x) = xa for all x $\in$ R except when $R\;{\cong}\;=M_2$(GF(2)) and I[I, I] = 0.

• East Asian mathematical journal
• /
• v.30 no.3
• /
• pp.259-270
• /
• 2014

For a generalized triangular matrix ring $$T=\[\array{R\;M\\0\;S}]$$, over rings R and S having only the idempotents 0 and 1 and over an (R, S)-bimodule M, we characterize all homomorphisms ${\alpha}$'s and all ${\alpha}$-derivations of T. Some of the homomorphisms are compositions of an inner homomorphism and an extended or a twisted homomorphism.

• Honam Mathematical Journal
• /
• v.31 no.4
• /
• pp.601-609
• /
• 2009

The notion of generalized derivations of a BCI-algebra is introduced, and some related properties are investigated. Also, the concept of a torsion free BCI-algebra is introduced and some properties are discussed.

• Communications of the Korean Mathematical Society
• /
• v.39 no.1
• /
• pp.1-10
• /
• 2024

In this paper we will demonstrate some results on a prime ring with involution by introducing two generalized derivations acting on symmetric and skew symmetric elements. This approach allows us to generalize some well known results. Furthermore, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.

• Communications of the Korean Mathematical Society
• /
• v.39 no.1
• /
• pp.79-91
• /
• 2024

This article examines the connection between 3-derivations and the commutativity of a prime ring R with an involution * that fulfills particular algebraic identities for symmetric and skew symmetric elements. In practice, certain well-known problems, such as the Herstein problem, have been studied in the setting of three derivations in involuted rings.

• Journal of applied mathematics & informatics
• /
• v.36 no.1_2
• /
• pp.81-92
• /
• 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.