A Generalization of Posner’s Theorem on Derivations in Rings (original) (raw)

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.

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).

On Generalized Derivations of Semiprime Rings

International Journal of Pure and Apllied Mathematics, 2014

Let F be a commuting generalized derivation, with associated derivation d, on a semiprime ring R. We show that d(x)[y, z] = 0 for all x, y, z ∈ R and d is central. We define and characterize dependent elements of F and investigate a decomposition of R relative to F .

Remarks on Semiprime Rings with Generalized Derivations

2012

Let R be a ring with centre Z(R). An additive mapping F : R −→ R is said to be a generalized derivation if there exists a derivation d : R −→ R such that F(xy )= F(x)y+xd(y), for all x,y ∈ R (the map d is called the derivation associated with F). In the present note we prove that if a semiprime ring R admits a generalized derivation F, d is the nonzero associated derivation of F, satisfying certain polynomial constraints on a nonzero ideal I, then R contains a nonzero central ideal. Mathematics Subject Classification: Primary 16N60; Secondary 16W25

A Note on Derivations of Commutative Rings

Proceedings of the 15th WSEAS international …, 2010

We study properties of a differentially simple commutative ring R with respect to a set D of derivations of R. Among the others we investigate the relation between the D-simplicity of R and that of the local ring R P with respect to a prime ideal P of R and we prove a criterion about the D-simplicity of R in case where R is a 1-dimensional (Krull dimension) finitely generated algebra over a field of characteristic zero and D is a singleton set. The above criterion was quoted without proof in an earlier paper of the author.

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.

On Algebraic Derivations of Prime Rings

Methods in Ring Theory, 1984

00-901 University of Warsaw 1. Preliminaries. It is well known that if R is a ring of characteristic p > 0 and d is a derivation of R, then d P is also a derivation. On the other hand, for a prime ring R, powers, less than char R, of inner derivations which are inner derivations were investigated in [3]. It appeared in particular that elements which determined such derivations have to be algebraic and the power of a derivation is not often a derivation. Our main objective is to extend the results of Martindale ([3J) to arbitrary algebraic• elements. Moreover, besides powers of derivations we wiil consider polynomials in derivations, providing natural assumptions on degree of polynomials and characteristic of a ring. Analogously as in [3] we will concentrate on prime rings only.

On (, )-Derivations in Prime Rings

2002

Let R be a 2-torsion free prime ring and let , be automor- phisms of R. For any x, y2 R, set (x,y) , = x(y) (y )x. Suppose that d is a (, )-derivati on defined on R. In the present paper it is shown that (i) if R satisfies (d( x),x) , = 0, then either d = 0 or R is commutative (ii) if I is a nonzero ideal of R suc h that (d(x),d (y)) = 0, for all x,y 2 I, and d commutes with both and , then either d = 0 or R is comm utative. (iii) if I is a nonzero ideal of R suc h that d(xy) = d( yx), for all x,y 2 I , and d commutes with , then R is comm utative. Finally a related result has been obtain for (, )-derivation.

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.