Action of prime ideals on generalized derivations-I (original) (raw)

On Ideals and Commutativity of Prime Rings with Generalized Derivations

European Journal of Pure and Applied Mathematics, 2018

An additive mapping F: R → R is called a generalized derivation on R if there exists a derivation d: R → R such that F(xy) = xF(y) + d(x)y holds for all x,y ∈ R. It is called a generalized (α,β)−derivation on R if there exists an (α,β)−derivation d: R → R such that the equation F(xy) = F(x)α(y)+β(x)d(y) holds for all x,y ∈ R. In the present paper, we investigate commutativity of a prime ring R, which satisfies certain differential identities on left ideals of R. Moreover some results on commutativity of rings with involutions that satisfy certain identities are proved.

A note on generalized derivations on prime rings

Arabian Journal of Mathematics

Let R be a prime ring with the extended centroid C and symmetric Martindale quotient ring Q s (R). In this paper we prove the following result. Let F : R → R be a generalized derivation associated with a non-zero derivation d on R and let h be an additive map of R such that F(x)x = xh(x) for all x ∈ R. Then either R is commutative or F(x) = x p and h(x) = px where p ∈ Q s (R).

A note on generalized derivations of prime rings

International Journal of Algebra, 2011

In the present paper we prove the following result; Let R be a noncommutative prime ring, I an ideal of R, (F, d) a generalized derivation of R and a ∈ R. If F ([x, a]) = 0 or [F (x), a] = 0 for all x ∈ I, then, d(x) = λ[x, a] for all x ∈ I or a ∈ Z.

Some Notes on Generalized P-Derivations with Ideals in Factor Rings

The main goal of this article is to delve deeper into the discussion of the commutativity of a factor ring R/P by analyzing certain differential identities that involve generalized P-derivations and P-multipliers connecting I to P. Here, I represents a non-zero ideal of an arbitrary ring R, and P is a prime ideal of R such that P ⊊ I. Furthermore, we will explore some outcomes from our various theorems. To underscore the necessity of the primeness assumption in our theorems, we will provide some illustrative counterexamples.

Generalized derivations in prime rings

International Journal of Mathematics Trends and Technology, 2017

Let R be a prime ring and I be a non zero ideal of R. Suppose that F, G, H : R → R are generalized derivations associated with derivations d, g, h respectively. If the following holds (i)F (xy)+G(x)H(y)+[α(x), y] = 0; for all x, y ∈ I, where α is any map on R, then R is commutative.

Specific Identities Involving Prime Ideals with Generalized P -derivations

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS, 2025

In this article, we will investigate the commutativity of the factor ring ℜ/P , where P is a prime ideal of any ring ℜ. This investigation will be carried out using generalized P-derivations ℧ and ⨿ associated with P-derivations χ and ∝, respectively, that satisfy specific functional identities linking ℜ to P. Moreover, we will discuss some related results. Finally, to reinforce the importance of our assumption regarding the primeness of P , we will provide some examples.

Notes on generalized derivations of *-prime rings

Miskolc Mathematical Notes, 2014

Let R be a -prime ring with characteristic different from two and U ¤ 0 be a square closed -Lie ideal of R. An additive mapping F W R ! R is called an generalized derivation if there exits a derivation d W R ! R such that F .xy/ D F .x/y C xd.y/. In the present paper, it is shown that U Â Z if R is a -prime ring which admits a generalized derivation satisfying several conditions that are associated with a derivation commuting with .

A note on multiplicative (generalized)-derivations in prime rings

International Journal of Algebra, 2016

Let R be a prime ring, 0 = a ∈ R and I a nonzero ideal of R. A map F : R → R is called a multiplicative (generalized)-derivation if F (xy) = F (x)y + xg(y) fulfilled for all x, y ∈ R where g : R → R is any map(not necessarily derivation). Suppose that F and G are two multiplicative (generalized)-derivations. The main objective of the present paper is to study the following situations: (i) a(G(xy)±[F (x), y]±yx) = 0; (ii) a(G(xy) + F (x)F (y) ± yx) = 0 for all x, y ∈ I.

A result on generalized derivations on right ideals of prime rings

Ukrainian Mathematical Journal, 2012

Let R be a prime ring of characteristic not 2 and let I be a nonzero right ideal of R. Let U be the right Utumi quotient ring of R and let C be the center of U. If G is a generalized derivation of R such that [[G(x), x], G(x)] = 0 for all x ∈ I, then R is commutative or there exist a, b ∈ U such that G(x) = ax + xb for all x ∈ R and one of the following assertions is true: (1) (a − λ)I = (0) = (b + λ)I for some λ ∈ C, (2) (a − λ)I = (0) for some λ ∈ C and b ∈ C. Нехай R-просте кiльце, характеристика якого не дорiвнює 2, а I-ненульовий правий iдеал R. Нехай Uправе фактор-кiльце Утумi кiльця R, а C-центр U. Якщо G є узагальненим диференцiюванням R таким, що [[G(x), x], G(x)] = 0 для всiх x ∈ I, то R є комутативним або iснують a, b ∈ U такi, що G(x) = ax + xb для всiх x ∈ R i виконується одне з наступних тверджень: (1) (a − λ)I = (0) = (b + λ)I для деякого λ ∈ C, (2) (a − λ)I = (0) для деяких λ ∈ C та b ∈ C. 1. Introduction. Throughout this paper R will always denote a prime ring with center Z(R), extended centroid C, right Utumi quotient ring U (sometimes, as in [2], U is called the maximal right ring of quotients), and two-sided Martindale quotient ring Q (see [2] for the definitions). For any x, y ∈ R, the commutator of x and y is denoted by [x, y] and defined to be xy − yx. An additive mapping d from R into itself is called a derivation of R if d(xy) = d(x)y + xd(y) holds for all x, y ∈ R. An additive mapping g : R → R is called a generalized derivation of R if there exists a derivation d of R such that g(xy) = g(x)y + xd(y) for all x, y ∈ R [10]. Obviously any derivation is a generalized derivation. Moreover, other basic examples of generalized derivations are the mappings of the form x → ax + xb, for a, b ∈ R. A generalized derivation in this form is called (generalized) inner. Many authors have studied generalized derivations in the context of prime and semiprime rings (see [1, 10, 13, 14]). In [13], T. K. Lee extended the definition of a generalized derivation as follows. By a generalized derivation he means an additive mapping g : I → U such that g(xy) = g(x)y +xd(y) for all x, y ∈ I, where I is a dense right ideal of the prime ring R and d is a derivation from I into U. He also proved that every generalized derivation can be uniquely extended to a generalized derivation of U, and moreover, there exist a ∈ U and a derivation d of U such that g(x) = ax + d(x) for all x ∈ U [13] (Theorem 3). In [7], De Filippis proved that if R is a prime ring of characteristic not 2 and G is a generalized derivation of R such that [[G(x), x], G(x)] = 0 for all x ∈ R, then either R is commutative or there exists λ ∈ C such that G(x) = λx for all x ∈ R. In the same paper, he uses his result to prove a theorem concerning noncommutative Banach algebras. More precisely, he proves the following:

ON COMMUTATIVITY OF *-PRIME RINGS WITH GENERALIZED DERIVATIONS

2013

Let R be an associative prime ring, U a Lie ideal such that u 2 ∈U for all u∈U and F: R→R be a generalized derivation. In this paper, we show that U ⊆ Z(R) if any one of the following conditions holds: (i) F(uv) -uv∈Z(R),(ii) F(uv) -vu∈Z(R), (iii) uF(v) + uv∈Z(R), and (iv) [F(u), v] + uv∈Z(R) for all u, v∈U. If we choose the underlying subset of R as an ideal instead of a Lie ideal of R, then we prove the commutativity of prime ring.