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Lie ideals with generalized derivations and derivations of semiprime rings

Sögütcü Emine Koç, Rehman Nadeem ur, Dernek Eda

Fasciculi Mathematici

2023

nr 66

107--120

Let R be a 2-torsion free semiprime ring, L a square-closed Lie ideal of R, ø be a derivation of R and α be an automorphism of R. We will demonstrate in this study that ø(L) = (0), and so ø is a zero map on L if any one of the following holds for all r r ∈ L: (i) ø(r)r = 0 (or r ø(r) = 0) (ii) ø(r) r + r (α (r) - r) = 0, (iii) The mapping r → ø (r) + α (r) is commuting on L. Moreover, if any one of the following are satisfied for two generalized derivations (F, ø) and (H, ξ) of R, then ø is a commuting map on L: (iv) F (r) F (s) - H(rs) ∈ Z (R), (v) F (rs) = ± H (rs), (vi) F (rs) = ±H (sr), for all r, s ∈ L.

Semiprime near-rings with multiplicative generalized (θ,θ)-derivations

Koç E., Gölbaşi Ö.

Fasciculi Mathematici

2016

Nr 57

105--119

Let N be a semiprime right near-ring and I a semigroup ideal of N. A map f : N →N is called a multiplicative generalized (θ,θ)-derivation if there exists a multiplicative (θ,θ)-derivation d : R → such that f(xy) - f(x)θ(y) + θ(x)d(y), for all x,y ϵ R. The purpose of this paper is to investigate the following: [formula].

Approximate generalized derivations close to derivations in Lie C*-algebras

Eshaghi M., Abbaszadeh S.

Journal of Applied Analysis

2015

Vol. 21, nr 1

37--43

We apply a fixed point theorem to prove that there exists a unique derivation close to an approximately generalized derivation in Lie C*-algebras. Also, we prove the hyperstability of generalized derivations. In other words, we find some conditions under which an approximately generalized derivation becomes a derivation.

Cocentralizing generalized derivations on multilinear polynomial on right ideals of prime rings

Filippis V. de, Dhara B.

Demonstratio Mathematica

2014

Vol. 47, nr 1

22--36

Let R be a prime ring with Utumi quotient ring U and with extended centroid C, I a non-zero right ideal of R, f(x1,…,xn) a multilinear polynomial over C which is not central valued on R and G, H two generalized derivations of R. Suppose that G(f(r))f(r) – f(r)H(f(r)) € C, for all r = (r1,…,rn) € In. Then one of the following holds: 1. there exist a; b; p (…) ; 2. R satisfies s4, the standard identity of degree 4, and there exist a; (…) ; 3. R satisfies s4 and there exist a; (…) ; 4. R satisfies s4 and there exist a; (…) ; 5. there exists e2 = e (…) and one of the following holds: (a) (…) is an identity for I; (b) char(R) = 2 and s4(x1, x2, x3, x4)x5 is an identity for I; (c) (…) is an identity for I and there exist a, a’, b, b’ (…) , a derivation of R, such that G(x) = ax + xa’ + d(x), H(x) = bx + xb’ – d(x), for all x (…) R, with (a – b’ – α)I = (0) = (b – a’ – α)I.

Identities with generalized derivations in semiprime rings

Dhara B., Ali S., Pattanayak A.

Demonstratio Mathematica

2013

Vol. 46, nr 3

453--460

Let R be a semiprime ring. An additive mapping F:R  R is called a generalized derivation of R if there exists a derivation d : R  R such that F(xy) = F(x)y + xd(y) holds, for all x,y  R. The objective of the present paper is to study the following situations: (1) (...), for all x, y in some appropriate subset of R.

Generalized finite operators

Mecheri S.

Demonstratio Mathematica

2005

Vol. 38, nr 1

163--167

Let B(H) be the algebra of all bounded linear operators on an infinite dimensional complex and separable Hilbert space H. A infinity B(H) is called finite if \\AX - XA - I\\ > 1, VX infinity B(H). In this paper we extend the class of finite operators to a more general class of pairs of operators called generalized finite operators defined by {(A, B) infinity B(H) x B(H) : \\AX - XB - I\\ > 1, VX infinity B(H)} and we present some pairs of generalized finite operators.

Finite operators

Mecheri S.

Demonstratio Mathematica

2002

Vol. 35, nr 2

357-366